A294947 -id:A294947 - cp's OEIS frontend

A294948 Expansion of Product_{n>=1} (1 - n^n*x^n)^(1/n).

1, -1, -2, -7, -57, -541, -7126, -108072, -1966034, -40620681, -952305757, -24824933859, -714742428220, -22491627743504, -768696164146118, -28344822040761041, -1121925480573229737, -47442205907345238412, -2134679753840086267669

Offset: 0

Seiichi Manyama, Nov 11 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/n, g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..387

Column k=0 of A294947.

Cf. A023881, A023887.

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

G.f.: exp(-Sum_{k>0} A023887(k)*x^k/k).

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A023887(k)*a(n-k) for n > 0.

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)).

A294951 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(-Sum_{j>=1} sigma_k(j) * x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, -3, 1, -1, -5, -1, 1, -1, -9, -7, 1, 1, -1, -17, -31, 1, 279, 1, -1, -33, -115, -23, 839, 301, 1, -1, -65, -391, -215, 3399, 4171, 12263, 1, -1, -129, -1267, -1319, 17519, 41311, 54305, 5601, 1, -1, -257, -3991, -6839, 102999, 387031, 473129, 102817, -431281

Offset: 0

Seiichi Manyama, Nov 12 2017

			Square array A(n,k) begins:
     1,   1,    1,     1,      1, ...
    -1,  -1,   -1,    -1,     -1, ...
    -3,  -5,   -9,   -17,    -33, ...
    -1,  -7,  -31,  -115,   -391, ...
     1,   1,  -23,  -215,  -1319, ...
   279, 839, 3399, 17519, 102999, ...

Columns k=0..2 give A294402, A294403, A294404.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000051(n+1).
Cf. A294947.

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

cp's OEIS Frontend

A294948 Expansion of Product_{n>=1} (1 - n^n*x^n)^(1/n).

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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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A294951 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(-Sum_{j>=1} sigma_k(j) * x^j).

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