A307063 Expansion of 1/(2 - Product_{k>=1} (1 + k*x^k)).

1, 1, 3, 10, 28, 85, 252, 745, 2202, 6530, 19326, 57194, 169341, 501242, 1483816, 4392531, 13002772, 38491212, 113943278, 337298400, 998482338, 2955742400, 8749688247, 25901125616, 76673399424, 226971213462, 671887935923, 1988945626648, 5887744768722, 17429103155892, 51594226501776

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• nonn

Comments

Invert transform of A022629.

a(n) is the number of compositions of n where there are A022629(k) sorts of part k. - Joerg Arndt, Jan 24 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A022629, A299164, A304969, A320652.

Cf. A307057, A307058, A307059, A307060, A307062.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+j*x^j): j in [1..m+2]])) ));

Mathematica

nmax = 30; CoefficientList[Series[1/(2 - Product[(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

SageMath

m=80;
def f(x): return 1/( 2 - product(1+j*x^j for j in range(1,m+3)) )
def A307063_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307063_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A022629(k)*a(n-k).