asm[n_,k_]:=If[k>0&&k≤n,Binomial[n+k-2,k-1](2n-k-1)!/(n-k)! Product[(3j+1)!/(n+j)!,{j,0,n-2}],0];

alpha[n_,x_]:=Sum[asm[n,k]x^(k-1),{k,1,n}];gamma[n_,x_]:=Sum[(-2(n-k+3)asm[n-1,k-3]+(5n-4k+6)asm[n-1,k-2]+(n+4k-6)asm[n-1,k-1]-2k asm[n-1,k] )x^(k-1),{k,1,n+2}];
delta[n_,x_]:=Sum[(4*(n+4-k)*(n+5-k)*asm[n-1,k-5]-4*(n+4-k)*(5*n+11-4*k)*asm[n-1,k-4]+(240-172*k+32*k^2+120*n-52*k*n+21*n^2)*asm[n-1,k-3]-2*(80-80*k+20*k^2+42*n-20*k*n-5*n^2)*asm[n-1,k-2]+(64-84*k+32*k^2-4*n-12*k*n+n^2)*asm[n-1,k-1]-4*k*(n-5+4*k)*asm[n-1,k]+4*k*(k+1)*asm[n-1,k+1])x^(k-1),{k,1,n+3}];

gammacoeff[n_,k_]:=-2(n-k+3)a[n-1,k-3]+(5n-4k+6)a[n-1,k-2]+(n+4k-6)a[n-1,k-1]-2k a[n-1,k] 
deltacoeff[n_,k_]:=4*(n+4-k)*(n+5-k)*a[n-1,k-5]-4*(n+4-k)*(5*n+11-4*k)*a[n-1,k-4]+(240-172*k+32*k^2+120*n-52*k*n+21*n^2)*a[n-1,k-3]-2*(80-80*k+20*k^2+42*n-20*k*n-5*n^2)*a[n-1,k-2]+(64-84*k+32*k^2-4*n-12*k*n+n^2)*a[n-1,k-1]-4*k*(n-5+4*k)*a[n-1,k]+4*k*(k+1)*a[n-1,k+1]

a[n_,k_]:=If[k<1,0,b[n,k]]

(*Proof by computer algebra for gamma*)

Table[gammacoeff[n,k]==c1( a[n,k]-2a[n,k-1]+a[n,k-2]) + c2( a[n-1,k-2]-a[n-1,k-1]) + c3 a[n-2,k-2],{k,1,3}]

Solve[%,{c1,c2,c3}]

Simplify[-2(n-k+3)b[n-1,k-3]+(5n-4k+6)b[n-1,k-2]+(n+4k-6)b[n-1,k-1]-2k b[n-1,k] -(-((2 b[-1+n,1])/(b[n,1]))( b[n,k]-2b[n,k-1]+b[n,k-2]) +(-(((2+n) b[-1+n,1]-4 b[-1+n,2])/(b[-1+n,1]))+((2 (2 b[n,1]-b[n,2]))/(b[n,1])))( b[n-1,k-2]-b[n-1,k-1]) + (((2 (-2 ((b[-1+n,1])^2)+3 n ((b[-1+n,1])^2)+2 ((b[-1+n,2])^2)-3 b[-1+n,1] b[-1+n,3]))/(b[-2+n,1] b[-1+n,1]))+((2 (-b[-1+n,1] b[n,1]+2 b[-1+n,2] b[n,1]-b[-1+n,1] b[n,2]-b[-1+n,2] b[n,2]+b[-1+n,1] b[n,3]))/(b[-2+n,1] b[n,1]))) b[n-2,k-2])/.{b[n_,k_]->Binomial[n+k-2,k-1](2n-k-1)!/(n-k)! Product[(3j+1)!/(n+j)!,{j,0,n-2}]}]

FullSimplify[(Binomial[-6+k+n,-3+k] (-1-k+n)! (1-k+n)! (2-k+n)! (-3 (-1+n) n ((((-3+n)!))^2) ((((-2+n)!))^2) (-6+2 n)! (-4+2 n)! (-2+2 n)!+(-4+n)! (4 (((-1+n))^2) ((((-2+n)!))^3) ((((-5+2 n)!))^2) (-2+2 n)!-2 ((((-3+n)!))^2) ((((-4+2 n)!))^2) (n (-1+n)! (-3+2 n)!-3 (-1+n) (-2+n)! (-2+2 n)!)+(-3+n)! (-2+n)! (-4+2 n)! (n (-1+n)! ((1+n) ((((-4+2 n)!))^2)-2 (-1+n) (-5+2 n)! (-3+2 n)!)+4 (-1+n) (-2+n)! (-5+2 n)! (-2+2 n)!))) (-3-k+2 n)!+2 (-4+n)! (-2+n)! (-6+2 n)! (k Binomial[-3+k+n,-1+k] (-3+n)! (-2+n)! (-k+n)! (1-k+n)! (2-k+n)! (-4+2 n)! (-2+2 n)! (-3-k+2 n)!+(-1-k+n)! (Binomial[-4+k+n,-2+k] (1-k+n)! (2-k+n)! (-2 (-1+n) ((((-2+n)!))^2) (-5+2 n)! (-2+2 n)!+(-3+n)! (-4+2 n)! (n (-1+n)! (-3+2 n)!-2 (-1+k) (-2+n)! (-2+2 n)!)) (-2-k+2 n)!-Binomial[-2+k+n,-1+k] (-3+n)! (-1+n)! (1-k+n)! (2-k+n)! ((((-4+2 n)!))^2) (-1-k+2 n)!-(-k+n)! (Binomial[-5+k+n,-3+k] (2-k+n)! (-2 (-1+n) ((((-2+n)!))^2) (-5+2 n)! (-2+2 n)!+(-3+n)! (-4+2 n)! (n (-1+n)! (-3+2 n)!+(2-2 k+3 n) (-2+n)! (-2+2 n)!)) (-1-k+2 n)!+(-3+n)! (-4+2 n)! (-2 Binomial[-3+k+n,-2+k] (-1+n)! (2-k+n)! (-4+2 n)! (-k+2 n)!+(1-k+n)! ((-3+k-n) Binomial[-6+k+n,-4+k] (-2+n)! (-2+2 n)! (-k+2 n)!+Binomial[-4+k+n,-3+k] (-1+n)! (-4+2 n)! (1-k+2 n)!))))))]

(*Proof by computer algebra for delta*)

Table[deltacoeff[n,k]==d1( -a[n,k]+3a[n,k-1]-3a[n,k-2]+a[n,k-3]) + d2( a[n-1,k-1]-2a[n-1,k-2]+a[n-1,k-3]) + d3( -a[n-2,k-2]+a[n-2,k-3])+d4 a[n-3,k-3],{k,1,4}]

Solve[%,{d1,d2,d3,d4}]

Simplify[4*(n+4-k)*(n+5-k)*b[n-1,k-5]-4*(n+4-k)*(5*n+11-4*k)*b[n-1,k-4]+(240-172*k+32*k^2+120*n-52*k*n+21*n^2)*b[n-1,k-3]-2*(80-80*k+20*k^2+42*n-20*k*n-5*n^2)*b[n-1,k-2]+(64-84*k+32*k^2-4*n-12*k*n+n^2)*b[n-1,k-1]-4*k*(n-5+4*k)*b[n-1,k]+4*k*(k+1)*b[n-1,k+1]-((-((-4 (-1+n) b[-1+n,1]+8 b[-1+n,2])/(b[n,1])))( -b[n,k]+3b[n,k-1]-3b[n,k-2]+b[n,k-3]) + ((((24-28 n+(n^2)) b[-1+n,1]-8 (3+n) b[-1+n,2]+24 b[-1+n,3])/(b[-1+n,1]))-(((-4 (-1+n) b[-1+n,1]+8 b[-1+n,2]) (-3 b[n,1]+b[n,2]))/(b[-1+n,1] b[n,1])))( b[n-1,k-1]-2b[n-1,k-2]+b[n-1,k-3]) + (-(((2 b[-1+n,1]-b[-1+n,2]) ((24-28 n+(n^2)) b[-1+n,1]-8 (3+n) b[-1+n,2]+24 b[-1+n,3]))/(b[-2+n,1] b[-1+n,1]))-((-2 (20-18 n-5 (n^2)) b[-1+n,1]+(100-40 n+(n^2)) b[-1+n,2]-12 (7+n) b[-1+n,3]+48 b[-1+n,4])/(b[-2+n,1]))+(1/(b[-2+n,1] b[-1+n,1] b[n,1]))(-4 (-1+n) b[-1+n,1]+8 b[-1+n,2]) (-3 b[-1+n,1] b[n,1]+3 b[-1+n,2] b[n,1]-b[-1+n,1] b[n,2]-b[-1+n,2] b[n,2]+b[-1+n,1] b[n,3]))( -b[n-2,k-2]+b[n-2,k-3])+(-(1/(b[-3+n,1] b[-2+n,1] b[-1+n,1]))((24-28 n+(n^2)) b[-1+n,1]-8 (3+n) b[-1+n,2]+24 b[-1+n,3]) (-b[-2+n,1] b[-1+n,1]+2 b[-2+n,2] b[-1+n,1]-b[-2+n,1] b[-1+n,2]-b[-2+n,2] b[-1+n,2]+b[-2+n,1] b[-1+n,3])-(((-b[-2+n,1]+b[-2+n,2]) (-2 (20-18 n-5 (n^2)) b[-1+n,1]+(100-40 n+(n^2)) b[-1+n,2]-12 (7+n) b[-1+n,3]+48 b[-1+n,4]))/(b[-3+n,1] b[-2+n,1]))+(((64-88 n+21 (n^2)) b[-1+n,1]-2 (80-38 n-5 (n^2)) b[-1+n,2]+(240-52 n+(n^2)) b[-1+n,3]-16 (11+n) b[-1+n,4]+80 b[-1+n,5])/(b[-3+n,1]))+(1/(b[n,1]))(-4 (-1+n) b[-1+n,1]+8 b[-1+n,2]) (1/(b[-3+n,1] b[-2+n,1] b[-1+n,1])(-b[-2+n,1] b[-1+n,1]+2 b[-2+n,2] b[-1+n,1]-b[-2+n,1] b[-1+n,2]-b[-2+n,2] b[-1+n,2]+b[-2+n,1] b[-1+n,3]) (-3 b[n,1]+b[n,2])+(((-b[-2+n,1]+b[-2+n,2]) (3 b[n,1]-3 b[n,2]+b[n,3]))/(b[-3+n,1] b[-2+n,1]))-((-b[n,1]+3 b[n,2]-3 b[n,3]+b[n,4])/(b[-3+n,1])))) b[n-3,k-3])/.{b[n_,k_]->Binomial[n+k-2,k-1](2n-k-1)!/(n-k)! Product[(3j+1)!/(n+j)!,{j,0,n-2}]}]

FullSimplify[(((12 k (1+k) Binomial[-2+k+n,k] (-4-k+2 n)!)/((-1+n)! (-2-k+n)!))-(2 (-1+n) Binomial[-8+k+n,-4+k] (5 n (2+3 n+(n^2)) (-5+n)! ((((-4+n)!))^2) ((((-3+n)!))^2) ((((-2+n)!))^2) (-8+2 n)! (-6+2 n)! (-4+2 n)! (-2+2 n)!+(-6+n)! (-4 n (1+n) (-4+n)! ((((-3+n)!))^2) ((((-2+n)!))^2) (-7+2 n)! (3 (-2+n) (-3+n)! (-7+2 n)!+(8+n) (-4+n)! (-6+2 n)!) (-4+2 n)! (-2+2 n)!+(-5+n)! (9 n ((((-3+n)!))^2) ((((-2+n)!))^2) (-6+2 n)! (-(-1+n) (-2+n)! (n ((((-6+2 n)!))^2)-2 (-2+n) (-7+2 n)! (-5+2 n)!)+(-6+n+(n^2)) (-3+n)! (-7+2 n)! (-4+2 n)!) (-2+2 n)!-6 ((((-4+n)!))^2) (-6+2 n)! (-n ((((-3+n)!))^2) (-1+n)! ((((-4+2 n)!))^2) (-3+2 n)!+2 (-1+n) n ((((-2+n)!))^3) ((((-5+2 n)!))^2) (-2+2 n)!+(-3+n)! (-2+n)! (-4+2 n)! (n (-1+n)! ((1+n) ((((-4+2 n)!))^2)-(-3+n) (-5+2 n)! (-3+2 n)!)+(13-8 n) (-3+n)! (-4+2 n)! (-2+2 n)!)+((((-2+n)!))^2) (-5+2 n)! (-2 n (-1+n)! ((1+n) ((((-4+2 n)!))^2)-(-1+n) (-5+2 n)! (-3+2 n)!)+(10+3 n-3 (n^2)) (-3+n)! (-4+2 n)! (-2+2 n)!))+(-4+n)! (-3+n)! (6 (-2+n) n ((((-3+n)!))^2) (-1+n)! (-7+2 n)! ((((-4+2 n)!))^2) (-3+2 n)!+6 (-1+n) n ((((-2+n)!))^3) (-5+2 n)! ((3+n) ((((-6+2 n)!))^2)-2 (-2+n) (-7+2 n)! (-5+2 n)!) (-2+2 n)!+(-3+n)! (-2+n)! (-4+2 n)! (n (-1+n)! ((2+3 n+(n^2)) (-6+2 n)! (-5+2 n)! (-4+2 n)!-3 (-1+n) n ((((-6+2 n)!))^2) (-3+2 n)!-3 (-2+n) (-7+2 n)! ((1+n) ((((-4+2 n)!))^2)-2 (-3+n) (-5+2 n)! (-3+2 n)!))-6 (10-11 n+3 (n^2)) (-3+n)! (-7+2 n)! (-4+2 n)! (-2+2 n)!)+2 ((((-2+n)!))^2) (-n (-1+n)! (-5+2 n)! ((2+3 n+(n^2)) (-6+2 n)! (-5+2 n)! (-4+2 n)!-3 (-1+n) n ((((-6+2 n)!))^2) (-3+2 n)!-3 (-2+n) (-7+2 n)! ((1+n) ((((-4+2 n)!))^2)-2 (-1+n) (-5+2 n)! (-3+2 n)!))+3 (-3+n)! (-3 (-6+n) n ((((-6+2 n)!))^2)+2 (10-9 n+2 (n^2)) (-7+2 n)! (-5+2 n)!) (-4+2 n)! (-2+2 n)!))))) (-4-k+2 n)!)/((-6+n)! (-5+n)! (-4+n)! ((((-3+n)!))^2) ((((-2+n)!))^2) (-1+n)! (-k+n)! (-8+2 n)! (-6+2 n)! (-4+2 n)! (-2+2 n)!)-((12 k (-5+4 k+n) Binomial[-3+k+n,-1+k] (-3-k+2 n)!)/((-1+n)! (-1-k+n)!))+((3 (64+32 (k^2)-4 n+(n^2)-12 k (7+n)) Binomial[-4+k+n,-2+k] (-2-k+2 n)!)/((-1+n)! (-k+n)!))+(6 (-1+n) (4 n (1+n) (-4+n)! ((((-3+n)!))^2) ((((-2+n)!))^2) (-7+2 n)! (-4+2 n)! (-2+2 n)!+(-5+n)! (-3 n (-3+n)! ((((-2+n)!))^2) (-6+2 n)! (2 (-1+n) (-2+n)! (-5+2 n)!+(3+n) (-3+n)! (-4+2 n)!) (-2+2 n)!+(-4+n)! (-2 n ((((-3+n)!))^2) (-1+n)! ((((-4+2 n)!))^2) (-3+2 n)!+4 (-1+n) n ((((-2+n)!))^3) ((((-5+2 n)!))^2) (-2+2 n)!+2 ((((-2+n)!))^2) (-5+2 n)! (-n (-1+n)! ((1+n) ((((-4+2 n)!))^2)-2 (-1+n) (-5+2 n)! (-3+2 n)!)+2 (5-2 n) (-3+n)! (-4+2 n)! (-2+2 n)!)+(-3+n)! (-2+n)! (-4+2 n)! (n (-1+n)! ((1+n) ((((-4+2 n)!))^2)-2 (-3+n) (-5+2 n)! (-3+2 n)!)+2 (-5+3 n) (-3+n)! (-4+2 n)! (-2+2 n)!)))) (-Binomial[-6+k+n,-3+k] (1-k+n)! (-3-k+2 n)!+Binomial[-7+k+n,-4+k] (-k+n)! (-2-k+2 n)!))/((-5+n)! (-4+n)! (-3+n)! ((((-2+n)!))^2) (-1+n)! (-k+n)! (1-k+n)! (-6+2 n)! (-4+2 n)! (-2+2 n)!)+((6 (-80-20 (k^2)-42 n+5 (n^2)+20 k (4+n)) Binomial[-5+k+n,-3+k] (-1-k+2 n)!)/((-1+n)! (1-k+n)!))+((3 (32 (k^2)-4 k (43+13 n)+3 (80+40 n+7 (n^2))) Binomial[-6+k+n,-4+k] (-k+2 n)!)/((-1+n)! (2-k+n)!))-(3 (12 (-1+n) n (-3+n)! ((((-2+n)!))^2) (-6+2 n)! (-2+2 n)!+(-4+n)! (4 (-1+n) n (-3+n)! (-1+n)! (-4+2 n)! (-3+2 n)!-8 (-1+n) n ((((-2+n)!))^2) (-5+2 n)! (-2+2 n)!+(-2+n)! (-8 (-1+n) n (-1+n)! (-5+2 n)! (-3+2 n)!+(36-40 n+(n^2)) (-3+n)! (-4+2 n)! (-2+2 n)!))) (Binomial[-4+k+n,-2+k] (1-k+n)! (2-k+n)! (-2-k+2 n)!+(-k+n)! (-2 Binomial[-5+k+n,-3+k] (2-k+n)! (-1-k+2 n)!+Binomial[-6+k+n,-4+k] (1-k+n)! (-k+2 n)!)))/((-4+n)! (-3+n)! (-2+n)! (-1+n)! (-k+n)! (1-k+n)! (2-k+n)! (-4+2 n)! (-2+2 n)!)-((12 (4-k+n) (11-4 k+5 n) Binomial[-7+k+n,-5+k] (1-k+2 n)!)/((-1+n)! (3-k+n)!))+((12 (4-k+n) (5-k+n) Binomial[-8+k+n,-6+k] (2-k+2 n)!)/((-1+n)! (4-k+n)!))+(12 (-1+n) (-2 (-2+n)! (-5+2 n)!+(-3+n)! (-4+2 n)!) (Binomial[-2+k+n,-1+k] (1-k+n)! (2-k+n)! (3-k+n)! (-1-k+2 n)!-(-k+n)! (3 Binomial[-3+k+n,-2+k] (2-k+n)! (3-k+n)! (-k+2 n)!+(1-k+n)! (-3 Binomial[-4+k+n,-3+k] (3-k+n)! (1-k+2 n)!+Binomial[-5+k+n,-4+k] (2-k+n)! (2-k+2 n)!))))/((-3+n)! (-2+n)! (-k+n)! (1-k+n)! (2-k+n)! (3-k+n)! (-2+2 n)!))]