A118349 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118346.
1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 30, 0, 1, 4, 18, 70, 201, 0, 1, 5, 26, 121, 487, 1445, 0, 1, 6, 35, 184, 873, 3592, 10900, 0, 1, 7, 45, 260, 1375, 6606, 27600, 85128, 0, 1, 8, 56, 350, 2010, 10672, 51728, 218566, 682505, 0, 1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0
Offset: 0
Examples
Show: T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1) at n=8,k=4: T(8,4) = T(7,4) - 2*T(7,3) + 2*T(8,3) + T(9,3) or: 1375 = 873 - 2*184 + 2*260 + 350. Triangle begins: 1; 1, 0; 1, 1, 0; 1, 2, 5, 0; 1, 3, 11, 30, 0; 1, 4, 18, 70, 201, 0; 1, 5, 26, 121, 487, 1445, 0; 1, 6, 35, 184, 873, 3592, 10900, 0; 1, 7, 45, 260, 1375, 6606, 27600, 85128, 0; 1, 8, 56, 350, 2010, 10672, 51728, 218566, 682505, 0; 1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Maple
T:= proc(n, k) option remember; if k<0 or k>n then 0; elif k=0 then 1; elif k=n then 0; else T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1); fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 17 2021 -
Mathematica
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -2*T[n-1, k-1] +2*T[n, k-1] +T[n+1, k-1] ]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2021 *) -
PARI
{T(n,k)=polcoeff((serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^k)))/2/(1-2*x))/x)^(n-k),k)} -
Sage
@CachedFunction def T(n, k): if (k<0 or k>n): return 0 elif (k==0): return 1 elif (k==n): return 0 else: return T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1) flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 17 2021
Formula
Since g.f. G=G(x) of A118346 satisfies: G = 1 - 2*x*G + 2*x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [3*T(n-1+j,k-j) - 2*T(n-2+j,k-j)] for n>k>=0.
Comments