A238439 Number of pairs (C,D) where C is a composition of u, D is a composition into distinct parts of v, and u + v = n.

1, 2, 4, 10, 20, 42, 90, 182, 370, 748, 1526, 3060, 6156, 12344, 24748, 49654, 99392, 198966, 398166, 796658, 1593694, 3188584, 6377714, 12756888, 25515312, 51033092, 102068728, 204141754, 408292220, 816590586, 1633192578, 3266399030, 6532817194, 13065657556

Offset: 0

Author

Joerg Arndt, Feb 27 2014

Keywords

• nonn

Comments

This is one possible "overcomposition" analog of overpartitions (see A015128), as overpartitions are pairs of partitions and partitions into distinct parts.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A236002.

Programs

Maple

c:= proc(n) c(n):= ceil(2^(n-1)) end:
b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
    expand(b(n, i-1)+`if`(i>n, 0, x*b(n-i, i-1))))) end:
d:= proc(n) d(n):= (p-> add(i!*coeff(p, x, i),
            i=0..degree(p)))(b(n$2)) end:
a:= proc(n) a(n):= add(c(i)*d(n-i), i=0..n) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 28 2014

Mathematica

With[{N=66}, s=((1-q)*Sum[q^(n*(n+1)/2)*n!/QPochhammer[q, q, n], {n, 0, N}] )/(1-2*q)+O[q]^N; CoefficientList[s, q]] (* Jean-François Alcover, Jan 17 2016, adapted from PARI *)

PARI

N=66;  q='q+O('q^N);
gfc=(1-q)/(1-2*q); \\ A011782
gfd=sum(n=0, N, n!*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ); \\ A032020
Vec( gfc * gfd )

Formula

G.f.: C(x) * D(x) where C(x) and D(x) are respectively g.f. of A011782 and A032020.

a(n) ~ c * 2^n, where c = 1.521048571756660822618351147397515199378647451699288... . - Vaclav Kotesovec, Apr 13 2017