A294583 -id:A294583 - cp's OEIS frontend

A294584 Expansion of Product_{k>=1} (1 - k^2*x^k)^(k^2).

1, -1, -16, -65, -79, 831, 4789, 15099, 6450, -162454, -922222, -2766802, -2885883, 17436735, 123724783, 429826690, 823702295, -482288111, -11124249861, -50746426204, -140257438535, -188609543978, 446818695084, 4049784546745, 15671762002156

Offset: 0

Seiichi Manyama, Nov 03 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..3799

Column k=2 of A294583.

Cf. A294586.

nmax = 30; CoefficientList[Series[Product[(1 - k^2*x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2017 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k^2*x^k)^k^2))

A294699 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(k*j) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -16, 0, 1, -1, -256, -713, 0, 1, -1, -4096, -531185, -64711, 1, 1, -1, -65536, -387416393, -4294405135, -9688521, 0, 1, -1, -1048576, -282429470945, -281474581032631, -95363000655153, -2165724176, 1, 1

Offset: 0

Seiichi Manyama, Nov 07 2017

			Square array begins:
    1,      1,           1,                1, ...
   -1,     -1,          -1,               -1, ...
   -1,    -16,        -256,            -4096, ...
    0,   -713,     -531185,       -387416393, ...
    0, -64711, -4294405135, -281474581032631, ...

Seiichi Manyama, Antidiagonals n = 0..38, flattened

Columns k=0..1 give A010815, A294704.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294583, A294653, A294756.

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

cp's OEIS Frontend

A294584 Expansion of Product_{k>=1} (1 - k^2*x^k)^(k^2).

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A294699 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(k*j) in powers of x.

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cp's OEIS Frontend

A294584 Expansion of Product_{k>=1} (1 - k^2*x^k)^(k^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A294699 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(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294699 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(k*j) in powers of x.