A294813 -id:A294813 - cp's OEIS frontend

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.

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.

A295245 Expansion of Product_{k>=1} 1/(1 + k^k*x^k)^k.

Original entry on oeis.org

1, -1, -7, -74, -902, -14075, -253551, -5307194, -124832925, -3278747898, -94780240390, -2995303153545, -102658540155454, -3794631664471440, -150460754913170964, -6371573878247136298, -287006135162339175131, -13703650554585427586271

Offset: 0

Seiichi Manyama, Nov 18 2017

sign

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

Cf. A266964, A294813, A295244.

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

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^(2+n)*(-1)^(n/d).

A294946 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 12, 1, 1, 9, 32, 82, 1, 1, 17, 90, 304, 725, 1, 1, 33, 260, 1162, 3537, 8811, 1, 1, 65, 762, 4516, 17435, 52010, 128340, 1, 1, 129, 2252, 17722, 86529, 310193, 895397, 2257687, 1, 1, 257, 6690, 69964, 431675, 1865766, 6286826, 18016416, 45658174

Offset: 0

Seiichi Manyama, Nov 11 2017

			Square array begins:
     1,    1,     1,     1,      1, ...
     1,    1,     1,     1,      1, ...
     3,    5,     9,    17,     33, ...
    12,   32,    90,   260,    762, ...
    82,  304,  1162,  4516,  17722, ...
   725, 3537, 17435, 86529, 431675, ...

Columns k=0..2 give A023881, A023882, A294813.
Rows n=0+1, 2 give A000012, A000051(n+1).
Cf. A294296, A294947.

G.f. of column k: Product_{j>0} 1/(1 - j^j*x^j)^(j^(k-1)).

cp's OEIS Frontend

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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A295245 Expansion of Product_{k>=1} 1/(1 + k^k*x^k)^k.

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A294946 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

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cp's OEIS Frontend

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.

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Formula

A295245 Expansion of Product_{k>=1} 1/(1 + k^k*x^k)^k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A294946 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

Views

Author

Keywords

Examples

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^(kj)x^j)^j in powers of x.