A294699 -id:A294699 - cp's OEIS frontend

A294704 Expansion of Product_{k>=1} (1 - k^k*x^k)^(k^k).

1, -1, -16, -713, -64711, -9688521, -2165724176, -675843638952, -280752881225790, -149800127712465769, -99844730464906330029, -81300082264515781043363, -79413710307214816810372248, -91652445696245266803423194130

Offset: 0

Seiichi Manyama, Nov 07 2017

sign

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

Column k=1 of A294699.

Cf. A294757, A294773.

nmax = 20; CoefficientList[Series[Product[(1 - k^k*x^k)^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294773(k)*a(n-k) for n > 0.

a(n) ~ -n^(2*n). - Vaclav Kotesovec, Nov 09 2017

A294756 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j(k*j) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 17, 3, 1, 1, 257, 746, 5, 1, 1, 4097, 531698, 66442, 7, 1, 1, 65537, 387424586, 4295533810, 9843731, 11, 1, 1, 1048577, 282429602018, 281475372654922, 95371863223331, 2187951485, 15, 1, 1, 16777217, 205891133143226, 18446744358295025890, 931322857677725493491, 4738477951108288824, 680615166718, 22

Offset: 0

Seiichi Manyama, Nov 08 2017

			Square array begins:
   1,     1,          1,               1, ...
   1,     1,          1,               1, ...
   2,    17,        257,            4097, ...
   3,   746,     531698,       387424586, ...
   5, 66442, 4295533810, 281475372654922, ...

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

Columns k=0..1 give A000041, A294757.
Rows n=0-1 give A000012.
Cf. A294699.

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

A294704 Expansion of Product_{k>=1} (1 - k^k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294756 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

cp's OEIS Frontend

A294704 Expansion of Product_{k>=1} (1 - k^k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294756 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(k*j)*x^j)^j(k*j) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294756 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j(k*j) in powers of x.