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A365441 Number of partitions of [2n] whose block maxima sum to 3n.

%I A365441 #71 Oct 03 2024 07:47:17
%S A365441 1,1,2,5,10,23,47,103,209,449,908,1909,3864,8011,16165,33244,66933,
%T A365441 136628,274876,558107,1121160,2268526,4552291,9183569,18417449,
%U A365441 37073504,74300048,149334422,299134695,600481001,1202436958,2411536369,4827532935,9675363921,19364235775
%N A365441 Number of partitions of [2n] whose block maxima sum to 3n.
%C A365441 Rows of A367955 and the reversed columns of A367955 converge to this sequence.
%H A365441 Alois P. Heinz, <a href="/A365441/b365441.txt">Table of n, a(n) for n = 0..3321</a>
%H A365441 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A365441 a(n) = A367955(2n,3n).
%e A365441 a(0) = 1: the empty partition.
%e A365441 a(1) = 1: 1|2.
%e A365441 a(2) = 2: 12|34, 134|2.
%e A365441 a(3) = 5: 123|456, 12456|3, 13|2456, 1456|23, 1|2|3456.
%e A365441 a(4) = 10: 1234|5678, 1235678|4, 124|35678, 125678|34, 134|25678, 135678|24, 14|235678, 15678|234, 1|23|45678, 1|245678|3.
%e A365441 a(5) = 23: 12345|6789(10), 12346789(10)|5, 1235|46789(10), 1236789(10)|45, 1245|36789(10), 1246789(10)|35, 125|346789(10), 126789(10)|345, 12|3|456789(10), 1345|26789(10), 1346789(10)|25, 135|246789(10), 136789(10)|245, 13|2|456789(10), 145|236789(10), 146789(10)|235, 15|2346789(10), 16789(10)|2345, 1|234|56789(10), 1|2356789(10)|4, 1456789(10)|2|3, 1|24|356789(10), 1|256789(10)|34.
%p A365441 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A365441       b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
%p A365441     end:
%p A365441 a:= n-> coeff(b(2*n, 0), x, 3*n):
%p A365441 seq(a(n), n=0..42);
%p A365441 # second Maple program:
%p A365441 b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
%p A365441       `if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
%p A365441        (t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
%p A365441     end:
%p A365441 a:= n-> b(3*n, 2*n, 0):
%p A365441 seq(a(n), n=0..42);
%t A365441 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 A365441 a[n_] := If[n == 0, 1, b[3n, 2n, 0]];
%t A365441 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Oct 03 2024, after _Alois P. Heinz_ *)
%Y A365441 Cf. A000110, A367955, A367969, A368204, A368675.
%K A365441 nonn
%O A365441 0,3
%A A365441 _Alois P. Heinz_, Dec 06 2023