A294808 -id:A294808 - cp's OEIS frontend

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

1, -1, -8, -73, -927, -13969, -254580, -5288596, -124795126, -3272571133, -94692028369, -2991756529687, -102571647087930, -3791499758414848, -150359326161180392, -6367668575791613601, -286854342016830115157, -13697147209040205869792

Offset: 0

Seiichi Manyama, Nov 09 2017

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

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

Column k=1 of A294808.

Cf. A294810.

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

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A294810(k)*a(n-k) for n > 0.

A294953 Expansion of Product_{k>=1} (1 - k^(2k)x^k)^k.

Original entry on oeis.org

1, -1, -32, -2155, -259701, -48496253, -13001952944, -4732375549802, -2246504006429898, -1348407213767476321, -998562531571744073815, -894380298523142455736017, -953030939828900988652689704, -1191547999931410291515116161158

Offset: 0

Seiichi Manyama, Nov 12 2017

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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..213

Column k=2 of A294808.

Cf. A294954, A294955.

N=20; x='x+O('x^N); Vec(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.

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.

cp's OEIS Frontend

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

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A294953 Expansion of Product_{k>=1} (1 - k^(2k)x^k)^k.

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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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cp's OEIS Frontend

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A294953 Expansion of Product_{k>=1} (1 - k^(2*k)*x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

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Examples

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Formula

A294953 Expansion of Product_{k>=1} (1 - k^(2k)x^k)^k.

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.