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A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.

%I A365903 #21 Oct 03 2024 07:52:57
%S A365903 1,1,1,2,4,10,29,101,367,1562,6891,37871,197930,1121634,6888085,
%T A365903 46190282,323250987,2349020516,17897285514,142512956148,1178963284732,
%U A365903 10248806222398,91421283039658,847666112839362,8100455404172267,79925567946537362,814508927747776069
%N A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.
%H A365903 Alois P. Heinz, <a href="/A365903/b365903.txt">Table of n, a(n) for n = 0..100</a>
%H A365903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A365903 b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
%p A365903      `if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
%p A365903       t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
%p A365903     end:
%p A365903 a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
%p A365903 seq(a(n), n=0..26);
%p A365903 # second Maple program:
%p A365903 a:= proc(h) option remember; local b; b:=
%p A365903       proc(n, m) option remember; `if`(n=0, 1,
%p A365903         b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
%p A365903       end: forget(b); max(coeffs(b(h, 0)))
%p A365903     end:
%p A365903 seq(a(n), n=0..26);
%t A365903 Q[1, t_, s_] := t*s;
%t A365903 Q[n_, t_, s_] := Q[n, t, s] = s*D[Q[n-1, t, s], s] + s*t^n*Q[n-1, t, s] // Expand;
%t A365903 P[n_, t_] := Module[{s}, Q[n, t, s] /. s -> 1];
%t A365903 a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[P[n, t], t] // Max]];
%t A365903 Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Oct 03 2024 *)
%Y A365903 Row maxima of A124327.
%Y A365903 Cf. A367969.
%K A365903 nonn
%O A365903 0,4
%A A365903 _Alois P. Heinz_, Dec 14 2023