A385063 - OEIS

A385063

Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A385060.

1, 1, 0, 1, 1, 0, 1, 2, 9, 0, 1, 3, 20, 43, 0, 1, 4, 33, 140, 125, 0, 1, 5, 48, 297, 1080, -6279, 0, 1, 6, 65, 520, 3189, -3568, -412025, 0, 1, 7, 84, 815, 6800, 18003, -828668, -9060911, 0, 1, 8, 105, 1188, 12285, 70464, -1033749, -25887384, -98234103, 0

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OFFSET

0,8

LINKS

Table of n, a(n) for n=0..54.

FORMULA

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

0, 1, 2, 3, 4, 5, ...

0, 9, 20, 33, 48, 65, ...

0, 43, 140, 297, 520, 815, ...

0, 125, 1080, 3189, 6800, 12285, ...

0, -6279, -3568, 18003, 70464, 168125, ...

PROG

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));

a(n, k) = b(n, -k);

CROSSREFS

Columns k=0..1 give A000007, A385060.

Sequence in context: A127558 A324330 A197294 * A384987 A395148 A228249

Adjacent sequences: A385060 A385061 A385062 * A385064 A385065 A385066

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Jun 16 2025

STATUS

approved