A307062 Expansion of 1/(2 - Product_{k>=1} (1 + x^k)^k).

1, 1, 3, 10, 29, 88, 264, 790, 2366, 7086, 21216, 63523, 190201, 569485, 1705121, 5105383, 15286247, 45769238, 137039743, 410316854, 1228548190, 3678451550, 11013817655, 32976968175, 98737827756, 295635383297, 885175234817, 2650343093602, 7935511791620, 23760073760720, 71141108467679

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• nonn

Comments

Invert transform of A026007.

a(n) is the number of compositions of n where there are A026007(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. A026007, A257674, A299167, A304969.

Cf. A307057, A307058, A307059, A307060, A307063.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[(1+x^j)^j: j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

Maple

b:= proc(n) b(n):= add((-1)^(n/d+1)*d^2, d=numtheory[divisors](n)) end:
g:= proc(n) g(n):= `if`(n=0, 1, add(b(k)*g(n-k), k=1..n)/n) end:
a:= proc(n) a(n):= `if`(n=0, 1, add(g(k)*a(n-k), k=1..n)) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 24 2024

Mathematica

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

SageMath

m=80;
def f(x): return 1/( 2 - product((1+x^j)^j for j in range(1,m+3)) )
def A307062_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307062_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A026007(k)*a(n-k).