A294954 -id:A294954 - cp's OEIS frontend

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

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

Offset: 0

Seiichi Manyama, Nov 12 2017

sign

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.

A294955 a(n) = Sum_{d|n} d^(2*n+2).

Original entry on oeis.org

1, 65, 6562, 1049601, 244140626, 78368963450, 33232930569602, 18014467229220865, 12157665462543713203, 10000002384185795209930, 9849732675807611094711842, 11447546167874515876354097130, 15502932802662396215269535105522

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..213

Cf. A294645, A294810, A294953, A294954.

PARI
```
{a(n) = sigma(n, 2*n+2)}
```

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

G.f.: Sum_{k>0} k^(2*k+2)*x^k/(1-(k^2*x)^k).

cp's OEIS Frontend

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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Formula

A294955 a(n) = Sum_{d|n} d^(2*n+2).

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Author

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PARI

PARI

Formula

cp's OEIS Frontend

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

Keywords

Examples

Links

Crossrefs

Formula

A294955 a(n) = Sum_{d|n} d^(2*n+2).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

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.