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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A262495.

Sequence in context: A319074 A195195 A141827 * A350608 A143077 A203011

Adjacent sequences: A262526 A262527 A262528 * A262530 A262531 A262532

Keyword

nonn

Author

Alois P. Heinz, Sep 24 2015

Status

approved