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A292747 Number of partitions of 2n with exactly n kinds of 1's which are introduced in ascending order.

%I A292747 #15 May 06 2025 11:13:22
%S A292747 1,1,8,97,1778,43747,1349703,50033463,2164920950,107074391802,
%T A292747 5957871478583,368330684797595,25046735249606820,1857906353180702199,
%U A292747 149289720057575358424,12917953683720554797237,1197556745092101849164899,118414507831659267311128558
%N A292747 Number of partitions of 2n with exactly n kinds of 1's which are introduced in ascending order.
%H A292747 Alois P. Heinz, <a href="/A292747/b292747.txt">Table of n, a(n) for n = 0..200</a>
%F A292747 a(n) = A292746(2n,n).
%F A292747 a(n) ~ 2^(2*n) * n^(n-1/2) / (sqrt(2*Pi*(1-c)) * exp(n) * c^n * (2-c)^n), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.40637573995995990767695812412483975821... - _Vaclav Kotesovec_, Sep 28 2017
%e A292747 a(2) = 8: 21a1b, 1a1a1a1b, 1a1a1b1a, 1a1a1b1b, 1a1b1a1a, 1a1b1a1b, 1a1b1b1a, 1a1b1b1b  (the two kinds of 1's are denoted by 1a and 1b).
%p A292747 f:= (n, k)-> add(Stirling2(n, j), j=0..k):
%p A292747 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292747       f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292747     end:
%p A292747 a:= n-> b(2*n$2, n)-b(2*n$2, n-1):
%p A292747 seq(a(n), n=0..20);
%t A292747 f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
%t A292747 b[n_, i_, k_] := b[n, i, k] = If[n==0 || i<2, f[n, k], Sum[b[n - i*j, i-1, k], {j, 0, n/i}]];
%t A292747 a[n_] := b[2n, 2n, n] - b[2n, 2n, n-1];
%t A292747 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y A292747 Cf. A292746.
%K A292747 nonn
%O A292747 0,3
%A A292747 _Alois P. Heinz_, Sep 22 2017