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A243149 Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.

%I A243149 #10 May 21 2018 08:04:53
%S A243149 1,1,0,0,4,3,4,0,11,31,70,177,242,382,482,874,1655,4440,10696,24390,
%T A243149 49867,95850,172980,289229,492233,811753,1468084,2813206,5929361,
%U A243149 12780690,27858421,59275097,122326098,243179349,467856049,873044584,1588187110,2842593612
%N A243149 Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.
%H A243149 Alois P. Heinz, <a href="/A243149/b243149.txt">Table of n, a(n) for n = 0..200</a>
%e A243149 a(8) = 11: [1,1,3,3], [1,3,1,3], [1,3,3,1], [3,1,1,3], [3,1,3,1], [3,3,1,1], [1,1,1,1,4], [1,1,1,4,1], [1,1,4,1,1], [1,4,1,1,1], [4,1,1,1,1].
%p A243149 b:= proc(n, i, p) option remember; `if`(n=0, p!,
%p A243149       `if`(i<1, 0, expand(add(x^`if`(j=0, 0, i-j)*
%p A243149        b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
%p A243149     end:
%p A243149 a:= n-> coeff(b(n$2, 0), x, 0):
%p A243149 seq(a(n), n=0..50);
%t A243149 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, Expand[Sum[x^If[j == 0, 0, i - j]*b[n - i*j, i - 1, p + j]/j!, {j,0, n/i}]]]];
%t A243149 a[n_] := Coefficient[b[n, n, 0], x, 0];
%t A243149 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%Y A243149 Cf. A114638 (the same for partitions).
%K A243149 nonn
%O A243149 0,5
%A A243149 _Alois P. Heinz_, May 30 2014