A361607 -id:A361607 - cp's OEIS frontend

A361600 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,kj)/j!.

1, 1, 2, 1, 2, 5, 1, 2, 7, 16, 1, 2, 9, 34, 65, 1, 2, 11, 58, 209, 326, 1, 2, 13, 88, 473, 1546, 1957, 1, 2, 15, 124, 881, 4626, 13327, 13700, 1, 2, 17, 166, 1457, 10526, 52537, 130922, 109601, 1, 2, 19, 214, 2225, 20326, 145867, 677594, 1441729, 986410

Offset: 0

Seiichi Manyama, Mar 17 2023

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    2,    2,    2,     2,     2,     2, ...
    5,    7,    9,    11,    13,    15, ...
   16,   34,   58,    88,   124,   166, ...
   65,  209,  473,   881,  1457,  2225, ...
  326, 1546, 4626, 10526, 20326, 35226, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000522, A002720, A361598, A361599.
Main diagonal gives A361607.
Cf. A293012.

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

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

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

A361617 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)*(k+1),n-k)/k!.

Original entry on oeis.org

1, 2, 15, 214, 4721, 146046, 5958367, 307382090, 19459587009, 1478414285146, 132440451881231, 13787717744245182, 1647673524863409265, 223671725058601427414, 34184743554559413628191, 5837132027535188545269106, 1106136052471647285563082497

Offset: 0

Seiichi Manyama, Mar 18 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..272

Main diagonal of A361616.

Cf. A361607.

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

a(n) = n! * [x^n] exp( x/(1-x)^n ) / (1-x)^n.

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

cp's OEIS Frontend

A361600 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,kj)/j!.

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A361617 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)*(k+1),n-k)/k!.

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cp's OEIS Frontend

A361600 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,k*j)/j!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A361617 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)*(k+1),n-k)/k!.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A361600 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,kj)/j!.