A294291 -id:A294291 - cp's OEIS frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

1, 1, 0, 1, -1, 0, 1, -1, 3, 0, 1, -1, 1, -13, 0, 1, -1, 1, -1, 73, 0, 1, -1, 1, -7, 25, -501, 0, 1, -1, 1, -7, 73, -241, 4051, 0, 1, -1, 1, -7, 49, -421, 1081, -37633, 0, 1, -1, 1, -7, 49, -181, 2641, -3361, 394353, 0, 1, -1, 1, -7, 49, -301, 1561, -32131, 68881

Offset: 0

Seiichi Manyama, Oct 27 2017

			Square array A(n,k) begins:
   1,    1,    1,    1,    1, ...
   0,   -1,   -1,   -1,   -1, ...
   0,    3,    1,    1,    1, ...
   0,  -13,   -1,   -7,   -7, ...
   0,   73,   25,   73,   49, ...
   0, -501, -241, -421, -181, ...

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

Columns k=0..5 give A000007, A293125, A294290, A294291, A294292, A294293.
Rows n=0 gives A000012.
Main diagonal gives A294261.
Cf. A294212.

B(j,k) is the coefficient of Product_{i=1..k} 1/(1+x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

A294251 E.g.f.: exp((1+x)(1+x^2)(1+x^3) - 1).

Original entry on oeis.org

1, 1, 3, 19, 97, 681, 5971, 50443, 512289, 5821777, 67968451, 880345731, 12230425153, 177123301369, 2759869352787, 45321758937211, 776008738668481, 14020831026652833, 264488668445049859, 5187662544182915827, 106241960755765408161, 2257485400533886011721

Offset: 0

Seiichi Manyama, Oct 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..484

Column k=3 of A294250.

Cf. A294214, A294291.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1+x)*(1+x^2)*(1+x^3)-1)))

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 4*(n-3)*(n-2)*(n-1)*a(n-4) + 5*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) + 6*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Dec 02 2021

cp's OEIS Frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294251 E.g.f.: exp((1+x)(1+x^2)(1+x^3) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294251 E.g.f.: exp((1+x)*(1+x^2)*(1+x^3) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A294251 E.g.f.: exp((1+x)(1+x^2)(1+x^3) - 1).