A294950 -id:A294950 - cp's OEIS frontend

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, 0, 1, -1, -128, -2155, -927, 4, 1, -1, -512, -58921, -259701, -13969, 4, 1, -1, -2048, -1593811, -67045719, -48496253, -254580, 7, 1, -1, -8192, -43044673, -17178209325, -152513227585, -13001952944, -5288596, 3

Offset: 0

Seiichi Manyama, Nov 09 2017

			Square array begins:
    1,      1,         1,             1,                1, ...
   -1,     -1,        -1,            -1,               -1, ...
   -2,     -8,       -32,          -128,             -512, ...
   -1,    -73,     -2155,        -58921,         -1593811, ...
    0,   -927,   -259701,     -67045719,     -17178209325, ...
    4, -13969, -48496253, -152513227585, -476819162106101, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A073592, A294809, A294953.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A004171.
Cf. A294653, A294950.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2k)x^k)^k.

Original entry on oeis.org

1, 1, 33, 2220, 265132, 49163241, 13121450895, 4762820449382, 2257130616816421, 1353302193751862072, 1001440612663683369940, 896481723303781965832069, 954894526385647926192875010, 1193519555165192704579377833814

Offset: 0

Seiichi Manyama, Nov 12 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n^(2*n).

Seiichi Manyama, Table of n, a(n) for n = 0..214

Column k=2 of A294950.

Cf. A294953, A294955.

nmax = 20; CoefficientList[Series[Product[1/(1 - k^(2*k)*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2017 *)

N=20; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k^(2*k)*x^k)^k))

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294955(k)*a(n-k) for n > 0.

a(n) ~ n^(2*n+1). - Vaclav Kotesovec, Nov 15 2017

cp's OEIS Frontend

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2k)x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2*k)*x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

A294954 Expansion of Product_{k>=1} 1/(1 - k^(2k)x^k)^k.