A384801 - OEIS

A384801

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 A213108.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 10, 0, 1, 4, 15, 38, 41, 0, 1, 5, 24, 90, 216, 76, 0, 1, 6, 35, 172, 633, 1162, -2183, 0, 1, 7, 48, 290, 1424, 4668, 2236, -54998, 0, 1, 8, 63, 450, 2745, 12724, 30177, -102282, -1045567, 0, 1, 9, 80, 658, 4776, 28300, 113080, 43914, -3135056, -15948296, 0

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OFFSET

0,8

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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

EXAMPLE

Square array begins:

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

0, 1, 2, 3, 4, 5, 6, ...

0, 3, 8, 15, 24, 35, 48, ...

0, 10, 38, 90, 172, 290, 450, ...

0, 41, 216, 633, 1424, 2745, 4776, ...

0, 76, 1162, 4668, 12724, 28300, 55326, ...

0, -2183, 2236, 30177, 113080, 302305, 675252, ...

PROG

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

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

CROSSREFS

Columns k=0..1 give A000007, A213108.

Sequence in context: A143325 A307910 A128888 * A384802 A380178 A384804

Adjacent sequences: A384798 A384799 A384800 * A384802 A384803 A384804

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Jun 10 2025

STATUS

approved