A294609 -id:A294609 - cp's OEIS frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, -1, 1, -1, -128, -2155, -919, 5, 1, -1, -512, -58921, -259477, -13977, 1, 1, -1, -2048, -1593811, -67041751, -48496477, -253640, 13, 1, -1, -8192, -43044673, -17178144301, -152513231553, -13001163543, -5290184, 4

Offset: 0

Seiichi Manyama, Nov 04 2017

			Square array begins:
    1,    1,       1,         1,            1, ...
   -1,   -1,      -1,        -1,           -1, ...
   -2,   -8,     -32,      -128,         -512, ...
   -1,  -73,   -2155,    -58921,     -1593811, ...
   -1, -919, -259477, -67041751, -17178144301, ...

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

Columns k=0..2 give A022661, A294606, A294607.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283675, A294609.

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.

A294950 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 in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 33, 90, 13, 1, 1, 129, 2220, 1162, 24, 1, 1, 513, 59178, 265132, 17435, 48, 1, 1, 2049, 1594836, 67180330, 49163241, 310193, 86, 1, 1, 8193, 43048770, 17181660628, 152662629227, 13121450895, 6286826, 160

Offset: 0

Seiichi Manyama, Nov 12 2017

			Square array begins:
    1,     1,        1,            1,               1, ...
    1,     1,        1,            1,               1, ...
    3,     9,       33,          129,             513, ...
    6,    90,     2220,        59178,         1594836, ...
   13,  1162,   265132,     67180330,     17181660628, ...
   24, 17435, 49163241, 152662629227, 476855156157129, ...

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

Columns k=0..2 give A000219, A294813, A294954.
Rows n=0+1, 2 give A000012, A087289.
Cf. A294609, A294758, A294808.

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.

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Original entry on oeis.org

1, 1, 9, 90, 1154, 17427, 309117, 6285102, 144603015, 3717580810, 105696353842, 3293810230381, 111651093529948, 4089889271054734, 160989247361249558, 6776381334102511286, 303712681809603918633, 14439887378431671417669

Offset: 0

Seiichi Manyama, Nov 04 2017

nonn

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

Column k=1 of A294609.

Cf. A294606, A294608.

nmax = 20; CoefficientList[Series[Product[1/(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(1/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.

A294611 Expansion of Product_{k>=1} 1/(1 - kx^k)^(k^(2k)).

Original entry on oeis.org

1, 1, 33, 2220, 264908, 49163017, 13120646934, 4762819155533, 2257121941722156, 1353302171994081060, 1001440370811165212942, 896481721940248699989226, 954894511643935287905899347, 1193519554843091905978411389666

Offset: 0

Seiichi Manyama, Nov 04 2017

nonn

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

Column k=2 of A294609.

Cf. A294607.

nmax = 20; CoefficientList[Series[Product[1/(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(1/prod(k=1, N, (1-k*x^k)^k^(2*k)))

Convolution inverse of A294607.

cp's OEIS Frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

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A294950 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 in powers of x.

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Formula

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Views

Author

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Links

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Mathematica

PARI

Formula

A294611 Expansion of Product_{k>=1} 1/(1 - kx^k)^(k^(2k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j*x^j)^(j^(k*j)) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294950 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 in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294611 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^(2*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

A294950 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 in powers of x.

A294611 Expansion of Product_{k>=1} 1/(1 - kx^k)^(k^(2k)).