cp's OEIS Frontend

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.

%I A238439 #18 Apr 13 2017 18:54:36
%S A238439 1,2,4,10,20,42,90,182,370,748,1526,3060,6156,12344,24748,49654,99392,
%T A238439 198966,398166,796658,1593694,3188584,6377714,12756888,25515312,
%U A238439 51033092,102068728,204141754,408292220,816590586,1633192578,3266399030,6532817194,13065657556
%N 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.
%C A238439 This is one possible "overcomposition" analog of overpartitions (see A015128), as overpartitions are pairs of partitions and partitions into distinct parts.
%H A238439 Alois P. Heinz, <a href="/A238439/b238439.txt">Table of n, a(n) for n = 0..1000</a>
%F A238439 G.f.: C(x) * D(x) where C(x) and D(x) are respectively g.f. of A011782 and A032020.
%F A238439 a(n) ~ c * 2^n, where c = 1.521048571756660822618351147397515199378647451699288... . - _Vaclav Kotesovec_, Apr 13 2017
%p A238439 c:= proc(n) c(n):= ceil(2^(n-1)) end:
%p A238439 b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
%p A238439     expand(b(n, i-1)+`if`(i>n, 0, x*b(n-i, i-1))))) end:
%p A238439 d:= proc(n) d(n):= (p-> add(i!*coeff(p, x, i),
%p A238439             i=0..degree(p)))(b(n$2)) end:
%p A238439 a:= proc(n) a(n):= add(c(i)*d(n-i), i=0..n) end:
%p A238439 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 28 2014
%t A238439 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 *)
%o A238439 (PARI)
%o A238439 N=66;  q='q+O('q^N);
%o A238439 gfc=(1-q)/(1-2*q); \\ A011782
%o A238439 gfd=sum(n=0, N, n!*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ); \\ A032020
%o A238439 Vec( gfc * gfd )
%Y A238439 Cf. A236002.
%K A238439 nonn
%O A238439 0,2
%A A238439 _Joerg Arndt_, Feb 27 2014