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A262529 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order such that sorts of adjacent parts are different.

%I A262529 #12 Oct 25 2018 04:11:24
%S A262529 1,1,4,31,464,10423,307123,11087757,471750268,23064505722,
%T A262529 1272685923725,78185947269685,5290601944971906,390900941750607195,
%U A262529 31309282176759170370,2701913799542547998709,249913023732255442857064,24663493072687443375499678
%N A262529 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order such that sorts of adjacent parts are different.
%H A262529 Alois P. Heinz, <a href="/A262529/b262529.txt">Table of n, a(n) for n = 0..300</a>
%F A262529 a(n) = A262495(2n,n).
%F A262529 a(n) ~ 2^(2*n-2) * (n-1)! / (Pi * sqrt(1-c) * c^(n-1) * (2-c)^n), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599076769581241... - _Vaclav Kotesovec_, Oct 25 2018
%e A262529 a(2) = 4: 3a1b, 2a2b, 2a1b1a, 1a1b1a1b.
%p A262529 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
%p A262529       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
%p A262529     end:
%p A262529 A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
%p A262529 a:= n-> add(A(2*n, n-i)*(-1)^i/(i!*(n-i)!), i=0..n):
%p A262529 seq(a(n), n=0..20);
%t A262529 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; a[n_] := Sum[A[2*n, n-i]*(-1)^i/(i!*(n-i)!), {i, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 07 2017, translated from Maple *)
%Y A262529 Cf. A262495.
%K A262529 nonn
%O A262529 0,3
%A A262529 _Alois P. Heinz_, Sep 24 2015