A361616 - OEIS

A361616

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*(j+1),n-j)/j!.

1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 15, 34, 1, 1, 5, 25, 103, 209, 1, 1, 6, 37, 214, 885, 1546, 1, 1, 7, 51, 373, 2293, 9051, 13327, 1, 1, 8, 67, 586, 4721, 29176, 106843, 130922, 1, 1, 9, 85, 859, 8481, 70981, 427189, 1425495, 1441729, 1

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OFFSET

0,5

LINKS

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

FORMULA

E.g.f. of column k: exp( x/(1-x)^k ) / (1-x)^k.

T(n,k) = Sum_{j=0..n} (n+(k-1)*(j+1))!/(k*j+k-1)! * binomial(n,j) for k > 0.

EXAMPLE

Square array begins:

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

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

1, 7, 15, 25, 37, 51, ...

1, 34, 103, 214, 373, 586, ...

1, 209, 885, 2293, 4721, 8481, ...

1, 1546 ,9051, 29176, 70981, 146046, ...

PROG

(PARI) T(n, k) = n! * sum(j=0, n, binomial(n+(k-1)*(j+1), n-j)/j!);

CROSSREFS

Columns k=0..3 give A000012, A002720, A343884, A351767.

Main diagonal gives A361617.

Cf. A293985, A361600.

Sequence in context: A074662 A025243 A352765 * A341014 A145085 A228904

Adjacent sequences: A361613 A361614 A361615 * A361617 A361618 A361619

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Mar 18 2023

STATUS

approved