A118354 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118351.

1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, 42, 0, 1, 4, 21, 96, 325, 0, 1, 5, 30, 163, 770, 2688, 0, 1, 6, 40, 244, 1353, 6530, 23286, 0, 1, 7, 51, 340, 2093, 11760, 57612, 208659, 0, 1, 8, 63, 452, 3010, 18636, 105681, 523446, 1918314, 0, 1, 9, 76, 581, 4125, 27441, 170580, 973953, 4864795, 17994264, 0

Offset

0,8

Comments

A118351 equals the central terms of pendular triangle A118350 and the lower diagonals of this triangle form the semi-diagonals of the triangle A118350.

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

Since g.f. G=G(x) of A118351 satisfies: G = 1 - 3*x*G + 3*x*G^2 + x*G^3 then

T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1).

Recurrence involving antidiagonals:

T(n,k) = T(n-1,k) + Sum_{j=1..k} [4*T(n-1+j,k-j) - 3*T(n-2+j,k-j)] for n>k>=0.

Example

Show: T(n,k) = T(n-1,k) - 3*T(n-1,k-1) + 3*T(n,k-1) + T(n+1,k-1)
at n=8,k=4: T(8,4) = T(7,4) - 3*T(7,3) + 3*T(8,3) + T(9,3)
or: 2093 = 1353 - 3*244 + 3*340 + 452.
Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  6,   0;
  1, 3, 13,  42,    0;
  1, 4, 21,  96,  325,     0;
  1, 5, 30, 163,  770,  2688,      0;
  1, 6, 40, 244, 1353,  6530,  23286,      0;
  1, 7, 51, 340, 2093, 11760,  57612, 208659,       0;
  1, 8, 63, 452, 3010, 18636, 105681, 523446, 1918314,        0;
  1, 9, 76, 581, 4125, 27441, 170580, 973953, 4864795, 17994264, 0; ...

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n, 0, T[n-1, k] -3*T[n-1, k-1] +3*T[n, k-1] +T[n+1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2021 *)

Prog

(PARI) {T(n, k)=polcoeff((serreverse(x*(1-3*x+sqrt((1-3*x)*(1-7*x)+x*O(x^k)))/2/(1-3*x))/x)^(n-k), k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(Sage)
@CachedFunction
def T(n, k):
    if (k==0): return 1
    elif (k==n): return 0
    else: return T(n-1, k) - 3*T(n-1, k-1) + 3*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, Feb 18 2021

Crossrefs

Cf. A118350, A118351, A118352, A118353.

Row sums: A151616.

Sequence in context: A330327 A039907 A072340 * A080730 A232178 A016590

Adjacent sequences: A118351 A118352 A118353 * A118355 A118356 A118357

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 26 2006

Status

approved