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A118354 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118351. 5
1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, 42, 0, 1, 4, 21, 96, 325, 0, 1, 5, 30, 153, 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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..61.

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; ...

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(""))

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

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Last modified January 22 13:41 EST 2020. Contains 331149 sequences. (Running on oeis4.)