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

0,3

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

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

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 *)

Crossrefs

Cf. A209664-A209673.

Main diagonal of A209666 and A261718.

Cf. A261783.

Sequence in context: A318580 A054910 A028562 * A340028 A365030 A180829

Adjacent sequences: A209665 A209666 A209667 * A209669 A209670 A209671

Keyword

nonn

Author

Wouter Meeussen, Mar 11 2012

Extensions

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

Status

approved