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A242434 Number of compositions of n in which each part p has multiplicity p.

%I A242434 #15 Jul 15 2020 15:14:29
%S A242434 1,1,0,0,1,3,0,0,0,1,4,0,0,10,60,0,1,5,0,0,15,105,0,0,0,36,286,0,0,
%T A242434 1281,12768,0,0,0,56,504,1,7,2520,27720,28,378,1260,0,0,7014,84000,0,
%U A242434 0,4621,83168,360360,210,2346,2522880,37837800,13860,180180,120,1320
%N A242434 Number of compositions of n in which each part p has multiplicity p.
%C A242434 a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}.
%H A242434 Alois P. Heinz, <a href="/A242434/b242434.txt">Table of n, a(n) for n = 0..4500</a>
%e A242434 a(0) = 1: the empty composition.
%e A242434 a(1) = 1: [1].
%e A242434 a(4) = 1: [2,2].
%e A242434 a(5) = 3: [1,2,2], [2,1,2], [2,2,1].
%e A242434 a(9) = 1: [3,3,3].
%e A242434 a(10) = 4: [1,3,3,3], [3,1,3,3], [3,3,1,3], [3,3,3,1].
%e A242434 a(13) = 10: [2,2,3,3,3], [2,3,2,3,3], [2,3,3,2,3], [2,3,3,3,2], [3,2,2,3,3], [3,2,3,2,3], [3,2,3,3,2], [3,3,2,2,3], [3,3,2,3,2], [3,3,3,2,2].
%p A242434 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A242434        b(n, i-1, p) +`if`(i^2>n, 0, b(n-i^2, i-1, p+i)/i!)))
%p A242434     end:
%p A242434 a:= n-> b(n, isqrt(n), 0):
%p A242434 seq(a(n), n=0..100);
%t A242434 b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[i^2 >n, 0, b[n-i^2, i-1, p+i]/i!]]]; a[n_] := b[n, Floor[Sqrt[n]], 0]; Table[ a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)
%Y A242434 Cf. A033461 (the same for partitions), A336269.
%K A242434 nonn,look
%O A242434 0,6
%A A242434 _Alois P. Heinz_, May 14 2014