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
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