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A336031 Number of compositions of n such that the set of parts and the set of multiplicities of parts are equal.

%I A336031 #14 Apr 13 2022 07:40:09
%S A336031 1,1,0,0,4,3,4,0,5,1,70,120,122,130,446,0,277,726,370,1064,13751,
%T A336031 38913,41272,81168,137014,84448,300642,490540,517806,341033,1467180,
%U A336031 425328,2403512,2916863,4455856,39855808,164203236,216675811,447273890,730795760,1154455982
%N A336031 Number of compositions of n such that the set of parts and the set of multiplicities of parts are equal.
%p A336031 b:= proc(n, i, p, f, g) option remember; `if`(n=0, `if`(f=g, p!, 0),
%p A336031       `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, f, {f[], i}),
%p A336031       `if`(j=0, g, {g[], j}))/j!, j=0..n/i)))
%p A336031     end:
%p A336031 a:= n-> b(n$2, 0, {}$2):
%p A336031 seq(a(n), n=0..32);
%t A336031 b[n_, i_, p_, f_, g_] := b[n, i, p, f, g] = If[n == 0, If[f == g, p!, 0],
%t A336031      If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j,
%t A336031      If[j == 0, f, Union@Append[f, i]],
%t A336031      If[j == 0, g, Union@Append[g, j]]]/j!, {j, 0, n/i}]]];
%t A336031 a[n_] := b[n, n, 0, {}, {}];
%t A336031 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 13 2022, after _Alois P. Heinz_ *)
%Y A336031 Cf. A114640, A336032.
%K A336031 nonn
%O A336031 0,5
%A A336031 _Alois P. Heinz_, Jul 07 2020