A294605 -id:A294605 - cp's OEIS frontend

A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-jx^j)^(j^(kj)) in powers of x.

1, 1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 33, 90, 14, 1, 1, 129, 2220, 1154, 25, 1, 1, 513, 59178, 264908, 17427, 56, 1, 1, 2049, 1594836, 67176362, 49163017, 309117, 97, 1, 1, 8193, 43048770, 17181595604, 152662625259, 13120646934, 6285102, 198

Offset: 0

Seiichi Manyama, Nov 04 2017

			Square array begins:
    1,    1,      1,        1,           1, ...
    1,    1,      1,        1,           1, ...
    3,    9,     33,      129,         513, ...
    6,   90,   2220,    59178,     1594836, ...
   14, 1154, 264908, 67176362, 17181595604, ...

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

Columns k=0..2 give A006906, A294610, A294611.
Rows n=0-1 give A000012.
Cf. A294605.

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

A294606 Expansion of Product_{k>=1} (1 - k*x^k)^(k^k).

Original entry on oeis.org

1, -1, -8, -73, -919, -13977, -253640, -5290184, -124681406, -3272865905, -94671665085, -2991846831531, -102566663464544, -3791541404744714, -150357943095635464, -6367699625807475503, -286854179220742344135, -13697182490105378305606

Offset: 0

Seiichi Manyama, Nov 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..385

Column k=1 of A294605.

Cf. A294608.

nmax = 20; CoefficientList[Series[Product[(1 - k*x^k)^(k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)

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

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294608(k) * a(n-k) for n > 0.

A294607 Expansion of Product_{k>=1} (1 - kx^k)^(k^(2k)).

Original entry on oeis.org

1, -1, -32, -2155, -259477, -48496477, -13001163543, -4732376829091, -2246495500625034, -1348407234549297356, -998562296547665810106, -894380299878766527522232, -953030926136814034550634393, -1191548000252369142490485950437

Offset: 0

Seiichi Manyama, Nov 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..213

Column k=2 of A294605.

nmax = 20; CoefficientList[Series[Product[(1 - k*x^k)^(k^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)

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

cp's OEIS Frontend

A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-jx^j)^(j^(kj)) in powers of x.

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A294606 Expansion of Product_{k>=1} (1 - k*x^k)^(k^k).

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Author

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PARI

Formula

A294607 Expansion of Product_{k>=1} (1 - kx^k)^(k^(2k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

cp's OEIS Frontend

A294609 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*x^j)^(j^(k*j)) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294606 Expansion of Product_{k>=1} (1 - k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294607 Expansion of Product_{k>=1} (1 - k*x^k)^(k^(2*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A294609 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-jx^j)^(j^(kj)) in powers of x.

A294607 Expansion of Product_{k>=1} (1 - kx^k)^(k^(2k)).