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

1, 1, 8, 97, 1778, 43747, 1349703, 50033463, 2164920950, 107074391802, 5957871478583, 368330684797595, 25046735249606820, 1857906353180702199, 149289720057575358424, 12917953683720554797237, 1197556745092101849164899, 118414507831659267311128558

Offset: 0

Author

Alois P. Heinz, Sep 22 2017

Keywords

• nonn

Examples

			a(2) = 8: 21a1b, 1a1a1a1b, 1a1a1b1a, 1a1a1b1b, 1a1b1a1a, 1a1b1a1b, 1a1b1b1a, 1a1b1b1b  (the two kinds of 1's are denoted by 1a and 1b).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A292746.

Programs

Maple

f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(2*n$2, n)-b(2*n$2, n-1):
seq(a(n), n=0..20);

Mathematica

f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
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}]];
a[n_] := b[2n, 2n, n] - b[2n, 2n, n-1];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

Formula

a(n) = A292746(2n,n).

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