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.

1, 1, 4, 31, 464, 10423, 307123, 11087757, 471750268, 23064505722, 1272685923725, 78185947269685, 5290601944971906, 390900941750607195, 31309282176759170370, 2701913799542547998709, 249913023732255442857064, 24663493072687443375499678

Offset: 0

Author

Alois P. Heinz, Sep 24 2015

Keywords

• nonn

Examples

			a(2) = 4: 3a1b, 2a2b, 2a1b1a, 1a1b1a1b.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A262495.

Programs

Maple

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

Mathematica

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 *)

Formula

a(n) = A262495(2n,n).

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