A294463 -id:A294463 - cp's OEIS frontend

A294462 Expansion of e.g.f. Product_{k>0} (1-k*x^k)^(-1/k).

1, 1, 4, 18, 132, 900, 10080, 93240, 1285200, 16526160, 264600000, 3950100000, 81280584000, 1401728328000, 30861115084800, 663835444272000, 16425316331424000, 380082583808928000, 10885891543502976000, 279441709690118976000, 8697410321979899520000

Offset: 0

Seiichi Manyama, Oct 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..424

Column k=1 of A294761.

Cf. A028342, A055225, A294463.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k*x^k)^(1/k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} A055225(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

A294616 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: Product_{j>0} (1-j^k*x^j)^(1/j).

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -2, 1, 1, -1, -4, 0, -1, 1, -1, -8, -6, -12, 41, 1, -1, -16, -30, -72, 180, -131, 1, -1, -32, -114, -360, 840, -1080, 1499, 1, -1, -64, -390, -1656, 4200, -8640, 15120, -4159, 1, -1, -128, -1266, -7272, 22440, -69120, 161280, -45360

Offset: 0

Seiichi Manyama, Nov 05 2017

			Square array begins:
    1,   1,   1,    1,     1,      1, ...
   -1,  -1,  -1,   -1,    -1,     -1, ...
   -1,  -2,  -4,   -8,   -16,    -32, ...
    1,   0,  -6,  -30,  -114,   -390, ...
   -1, -12, -72, -360, -1656,  -7272, ...
   41, 180, 840, 4200, 22440, 126600, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A028343, A294463.
Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079, (-1)*A245804.
Cf. A294761.

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

A294465 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(-1/k).

Original entry on oeis.org

1, -1, 0, -6, 36, -180, 720, -7560, 236880, -3099600, 15120000, -194594400, 9989179200, -131935003200, 337815878400, -50154760656000, 2018231927712000, -27611162875296000, 363290246871552000, -12648028196067264000, 521752941995725440000

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Cf. A076717, A294356, A294462, A294463, A294464.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1+k*x^k)^(1/k))))

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

a(0) = 1 and a(n) = -(n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025

cp's OEIS Frontend

A294462 Expansion of e.g.f. Product_{k>0} (1-k*x^k)^(-1/k).

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A294616 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: Product_{j>0} (1-j^k*x^j)^(1/j).

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A294465 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(-1/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula