A294292 -id:A294292 - cp's OEIS frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

1, 1, 0, 1, -1, 0, 1, -1, 3, 0, 1, -1, 1, -13, 0, 1, -1, 1, -1, 73, 0, 1, -1, 1, -7, 25, -501, 0, 1, -1, 1, -7, 73, -241, 4051, 0, 1, -1, 1, -7, 49, -421, 1081, -37633, 0, 1, -1, 1, -7, 49, -181, 2641, -3361, 394353, 0, 1, -1, 1, -7, 49, -301, 1561, -32131, 68881

Offset: 0

Seiichi Manyama, Oct 27 2017

			Square array A(n,k) begins:
   1,    1,    1,    1,    1, ...
   0,   -1,   -1,   -1,   -1, ...
   0,    3,    1,    1,    1, ...
   0,  -13,   -1,   -7,   -7, ...
   0,   73,   25,   73,   49, ...
   0, -501, -241, -421, -181, ...

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

Columns k=0..5 give A000007, A293125, A294290, A294291, A294292, A294293.
Rows n=0 gives A000012.
Main diagonal gives A294261.
Cf. A294212.

B(j,k) is the coefficient of Product_{i=1..k} 1/(1+x^i).

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

A294252 E.g.f.: exp((1+x)(1+x^2)(1+x^3)*(1+x^4) - 1).

Original entry on oeis.org

1, 1, 3, 19, 121, 921, 8491, 89083, 1004529, 12855601, 180798931, 2688972771, 43446800233, 753263746249, 13821916933371, 268513532534731, 5515341294313441, 119063602254649953, 2692597826408668579, 63786073308011409331, 1576811081112158751321

Offset: 0

Seiichi Manyama, Oct 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..458

Column k=4 of A294250.

Cf. A294215, A294292.

N=66; x='x+O('x^N); Vec(serlaplace(exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4)-1)))

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 6*(n-2)*(n-1)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*a(n-4) + 10*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) + 12*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6) + 14*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7) + 8*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-8) + 9*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-9) + 10*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-10). - Vaclav Kotesovec, Dec 02 2021

cp's OEIS Frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

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A294252 E.g.f.: exp((1+x)(1+x^2)(1+x^3)*(1+x^4) - 1).

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cp's OEIS Frontend

A294289 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1+x^j) - 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294252 E.g.f.: exp((1+x)*(1+x^2)*(1+x^3)*(1+x^4) - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A294252 E.g.f.: exp((1+x)(1+x^2)(1+x^3)*(1+x^4) - 1).