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A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.

%I A258467 #21 Sep 28 2018 05:38:50
%S A258467 1,2,12,130,2216,52078,1558219,56524414,2406802476,117575627562,
%T A258467 6478447651345,397345158550386,26842747368209994,1980156804133210116,
%U A258467 158365138356099680582,13647670818304698139989,1260732993182758276252088,124273946254095006307105363
%N A258467 Number of partitions of 2n into parts of exactly n sorts which are introduced in ascending order.
%H A258467 Alois P. Heinz, <a href="/A258467/b258467.txt">Table of n, a(n) for n = 0..300</a>
%F A258467 a(n) = A256130(2n,n).
%F A258467 a(n) ~ 2^(2*n-1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 31 2015
%F A258467 a(n) ~ Stirling2(2*n, n) = A007820(n). - _Vaclav Kotesovec_, Jun 01 2015
%p A258467 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A258467       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A258467     end:
%p A258467 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p A258467 a:= n-> T(2*n, n):
%p A258467 seq(a(n), n=0..20);
%t A258467 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := T[2n, n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 06 2017, translated from Maple *)
%Y A258467 Cf. A256130, A187655, A187657, A217899, A217900, A245109, A319732.
%K A258467 nonn
%O A258467 0,2
%A A258467 _Alois P. Heinz_, May 30 2015