%I #14 Apr 13 2022 07:40:09

%S 1,1,0,0,4,3,4,0,5,1,70,120,122,130,446,0,277,726,370,1064,13751,

%T 38913,41272,81168,137014,84448,300642,490540,517806,341033,1467180,

%U 425328,2403512,2916863,4455856,39855808,164203236,216675811,447273890,730795760,1154455982

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

%p b:= proc(n, i, p, f, g) option remember; `if`(n=0, `if`(f=g, p!, 0),

%p `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, f, {f[], i}),

%p `if`(j=0, g, {g[], j}))/j!, j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0, {}$2):

%p seq(a(n), n=0..32);

%t b[n_, i_, p_, f_, g_] := b[n, i, p, f, g] = If[n == 0, If[f == g, p!, 0],

%t If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j,

%t If[j == 0, f, Union@Append[f, i]],

%t If[j == 0, g, Union@Append[g, j]]]/j!, {j, 0, n/i}]]];

%t a[n_] := b[n, n, 0, {}, {}];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 13 2022, after _Alois P. Heinz_ *)

%Y Cf. A114640, A336032.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jul 07 2020