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A360971 Number of multisets of size n with elements from [n] whose element sum is larger than the product of all elements.

%I A360971 #16 Dec 09 2023 09:00:16
%S A360971 0,0,2,4,6,8,12,14,17,20,23,26,30,33,38,41,44,47,52,55,60,63,68,71,76,
%T A360971 78,84,89,93,97,103,106,111,115,121,124,128,131,138,142,146,151,159,
%U A360971 162,168,171,176,181,187,190,196,201,206,210,218,221,227,232,238
%N A360971 Number of multisets of size n with elements from [n] whose element sum is larger than the product of all elements.
%H A360971 Alois P. Heinz, <a href="/A360971/b360971.txt">Table of n, a(n) for n = 0..400</a>
%e A360971 a(2) = 2: [1,1], [1,2].
%e A360971 a(3) = 4: [1,1,1], [1,1,2], [1,2,2], [1,1,3].
%e A360971 a(4) = 6: [1,1,1,1], [1,1,1,2], [1,1,2,2], [1,1,1,3], [1,1,2,3], [1,1,1,4].
%e A360971 a(8) = 17: [1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,2], [1,1,1,1,1,1,2,2], [1,1,1,1,1,2,2,2], [1,1,1,1,1,1,1,3], [1,1,1,1,1,1,2,3], [1,1,1,1,1,1,3,3], [1,1,1,1,1,1,1,4], [1,1,1,1,1,1,2,4], [1,1,1,1,1,1,3,4], [1,1,1,1,1,1,1,5], [1,1,1,1,1,1,2,5], [1,1,1,1,1,1,1,6], [1,1,1,1,1,1,2,6], [1,1,1,1,1,1,1,7], [1,1,1,1,1,1,2,7], [1,1,1,1,1,1,1,8].
%p A360971 b:= proc(n, i, s, p) `if`(s+n*i<=p, 0, `if`(n=0 or i=1, 1, g(n, i, s, p))) end:
%p A360971 g:= proc(n, i, s, p) option remember; add(b(n-1, j, s+j, p*j), j=1..i) end:
%p A360971 a:= n-> b(n$2, 0, 1):
%p A360971 seq(a(n), n=0..60);
%t A360971 b[n_, i_, s_, p_] := If[s + n*i <= p, 0, If[n == 0 || i == 1, 1, g[n, i, s, p]]];
%t A360971 g[n_, i_, s_, p_] := g[n, i, s, p] = Sum[b[n-1, j, s+j, p*j], {j, 1, i}];
%t A360971 a[n_] := b[n, n, 0, 1];
%t A360971 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 09 2023, after _Alois P. Heinz_ *)
%Y A360971 Cf. A103586, A360538, A360563.
%K A360971 nonn
%O A360971 0,3
%A A360971 _Alois P. Heinz_, Feb 27 2023