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

0,2

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

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

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 *)

Prog

(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 )

Crossrefs

Cf. A236002.

Sequence in context: A167193 A026666 A325508 * A121880 A094536 A323445

Adjacent sequences: A238436 A238437 A238438 * A238440 A238441 A238442

Keyword

nonn

Author

Joerg Arndt, Feb 27 2014

Status

approved