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A373306 Sum over all complete compositions of n of the element multiset size.

%I A373306 #23 Jun 23 2024 10:33:37
%S A373306 0,1,2,7,13,30,73,157,345,743,1650,3517,7593,16120,34294,72683,153475,
%T A373306 323293,679231,1423721,2977692,6218395,12959249,26970243,56037071,
%U A373306 116280086,240953162,498719275,1031029386,2129266321,4392871427,9054428894,18645998093
%N A373306 Sum over all complete compositions of n of the element multiset size.
%C A373306 A complete composition of n has element set [k] with k<=n (without gaps).
%H A373306 Alois P. Heinz, <a href="/A373306/b373306.txt">Table of n, a(n) for n = 0..1000</a>
%F A373306 G.f.: Sum_{k>0} d/dy C({1..k},x,y)|y = 1 where C({s},x,y) = Sum_{i in {s}} (C({s}-{i},x,y)*y*x^i)/(1 - Sum_{i in {s}} (y*x^i)) with C({},x,y) = 1. - _John Tyler Rascoe_, Jun 18 2024
%e A373306 a(1) = 1: 1.
%e A373306 a(2) = 2: 11.
%e A373306 a(3) = 7 = 2 + 2 + 3: 12, 21, 111.
%e A373306 a(4) = 13 = 3 + 3 + 3 + 4: 112, 121, 211, 1111.
%e A373306 a(5) = 30 = 3*3 + 4*4 + 5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111.
%p A373306 b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, [t!, 0], 0),
%p A373306      `if`(i<1 or n<i*(i+1)/2, 0, add((p-> p+[0, p[1]]*j)(
%p A373306         b(n-i*j, i-1, t+j)/j!), j=1..n/i)))
%p A373306     end:
%p A373306 a:= n-> add(b(n, k, 0)[2], k=0..floor((sqrt(1+8*n)-1)/2)):
%p A373306 seq(a(n), n=0..32);
%t A373306 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, {t!, 0}, {0, 0}], If[i < 1 || n < i*(i + 1)/2, {0, 0}, Sum[Function[p, p + {0, p[[1]]}*j][b[n - i*j, i - 1, t + j]/j!], {j, 1, n/i}]]];
%t A373306 a[n_] := Sum[b[n, k, 0][[2]], {k, 0, Floor[(Sqrt[1 + 8*n] - 1)/2]}];
%t A373306 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Jun 08 2024, after _Alois P. Heinz_ *)
%Y A373306 Cf. A107429, A373118, A373305.
%K A373306 nonn
%O A373306 0,3
%A A373306 _Alois P. Heinz_, May 31 2024