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A322884 Number of set partitions of [2n] such that the maximal absolute difference between the least elements of consecutive blocks equals n.

%I A322884 #15 Jan 06 2019 13:32:24
%S A322884 1,1,5,39,493,9320,242366,8193031,346270455,17780116911,1085004090887,
%T A322884 77324278953174,6344818280326312,592415284729545433,
%U A322884 62319734032202722887,7323734663214254662683,954467851066831095051393,137065739258353347820981920
%N A322884 Number of set partitions of [2n] such that the maximal absolute difference between the least elements of consecutive blocks equals n.
%C A322884 a(0) = 1 by convention.
%H A322884 Alois P. Heinz, <a href="/A322884/b322884.txt">Table of n, a(n) for n = 0..130</a>
%H A322884 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A322884 a(n) = A287215(2n,n).
%e A322884 a(1) = 1: 1|2.
%e A322884 a(2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.
%p A322884 b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
%p A322884      `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
%p A322884     end:
%p A322884 A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
%p A322884 a:= n-> A(2*n, n)-`if`(n=0, 0, A(2*n, n-1)):
%p A322884 seq(a(n), n=0..20);
%t A322884 b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m b[n - 1, k, m, l]];
%t A322884 A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
%t A322884 a[n_] := A[2 n, n] - If[n == 0, 0, A[2 n, n - 1]];
%t A322884 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jan 03 2019, translated from Maple *)
%Y A322884 Cf. A287215.
%K A322884 nonn
%O A322884 0,3
%A A322884 _Alois P. Heinz_, Dec 29 2018