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Number of compositions of n such that the set of parts and the set of multiplicities of parts are disjoint.
2

%I #19 Apr 13 2022 07:40:13

%S 1,0,2,2,2,4,6,6,14,34,79,159,227,429,727,1146,1999,3238,5018,8976,

%T 14977,24768,38400,70678,152535,295493,617675,1404099,3023086,6685876,

%U 14230031,30218806,62175519,127820798,257285277,516574751,1021334631,2009999405,3917878730

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

%p b:= proc(n, i, p, f, g) option remember; `if`(f intersect g<>{}, 0,

%p `if`(n=0, p!, `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0,

%p f, {f[], i}), `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[f ~Intersection~ g != {}, 0,

%t If[n == 0, p!, 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, 38}] (* _Jean-François Alcover_, Apr 13 2022, after _Alois P. Heinz_ *)

%Y Cf. A114639, A336031.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 07 2020