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A368102 Total number of partitions of [n-s] whose block maxima sum to s, summed over all s.

%I A368102 #18 Oct 03 2024 07:46:30
%S A368102 1,0,1,0,1,1,1,1,3,2,3,6,5,9,14,13,20,40,31,56,86,105,127,227,244,394,
%T A368102 520,665,911,1536,1565,2449,3507,4719,5931,9061,11151,16911,21774,
%U A368102 29798,39804,60411,71865,104399,144999,197907,253430,370044,478764,694807
%N A368102 Total number of partitions of [n-s] whose block maxima sum to s, summed over all s.
%H A368102 Alois P. Heinz, <a href="/A368102/b368102.txt">Table of n, a(n) for n = 0..5000</a>
%H A368102 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A368102 a(11) = 6: 123|4, 124|3, 13|24, 14|23, 1|2|34, 1|2345.
%p A368102 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A368102       b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
%p A368102     end:
%p A368102 a:= n-> add(coeff(b(n-k, 0), x, k), k=ceil(n/2)..n):
%p A368102 seq(a(n), n=0..80);
%p A368102 # second Maple program:
%p A368102 b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
%p A368102       `if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
%p A368102       (t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
%p A368102     end:
%p A368102 a:= n-> add(b(k, n-k, 0), k=ceil(n/2)..n):
%p A368102 seq(a(n), n=0..80);
%t A368102 b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
%t A368102 a[n_] := If[n == 0, 1, Sum[b[k, n - k, 0], {k, Ceiling[n/2], n}]];
%t A368102 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Oct 03 2024, after _Alois P. Heinz_ *)
%Y A368102 Antidiagonal sums of A367955.
%Y A368102 Cf. A365821.
%K A368102 nonn
%O A368102 0,9
%A A368102 _Alois P. Heinz_, Dec 11 2023