A209668 a(n) = count of monomials, of degree k = n, in the complete homogeneous symmetric polynomials h(mu,k) summed over all partitions mu of n.

1, 1, 7, 55, 631, 8001, 130453, 2323483, 48916087, 1129559068, 29442232007, 835245785452, 26113646252773, 880685234758941, 32191922753658129, 1259701078978200555, 52802268925363689079, 2352843030410455053891, 111343906794849929711260, 5567596199767400904172045

Offset: 0

Author

Wouter Meeussen, Mar 11 2012

Keywords

• nonn

Comments

a(n) is the number of partitions of n where each part i is marked with a word of length i over an n-ary alphabet whose letters appear in alphabetical order. a(2) = 7: 2aa, 2ab, 2bb, 1a1a, 1a1b, 1b1a, 1b1b. - Alois P. Heinz, Aug 30 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• Wikipedia, Symmetric Polynomials

Crossrefs

Cf. A209664-A209673.

Main diagonal of A209666 and A261718.

Cf. A261783.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

Mathematica

h[n_, v_] := Tr@ Apply[Times, Table[Subscript[x, j], {j, v}]^# & /@ Compositions[n, v], {1}]; h[par_?PartitionQ, v_] := Times @@ (h[#, v] & /@ par); Tr /@ Table[(h[#, l] & /@ Partitions[l]) /. Subscript[x, _] -> 1, {l, 10}]
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k] * Binomial[i+k-1, k-1]^j, {j, 0, n/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

Formula

a(n) ~ c * n^n, where c = A247551 = Product_{k>=2} 1/(1-1/k!) = 2.529477472... . - Vaclav Kotesovec, Nov 15 2016

a(n) = [x^n] Product_{k>=1} 1 / (1 - binomial(k+n-1,n-1)*x^k). - Ilya Gutkovskiy, May 09 2021

Extensions

a(0)=1 prepended and a(11)-a(19) added by Alois P. Heinz, Aug 29 2015