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A209668 a(n) = count of monomials, of degree k = n, in the complete homgeneous symmetric polynomials h(mu,k) summed over all partitions mu of n. 7
1, 1, 7, 55, 631, 8001, 130453, 2323483, 48916087, 1129559068, 29442232007, 835245785452, 26113646252773, 880685234758941, 32191922753658129, 1259701078978200555, 52802268925363689079, 2352843030410455053891, 111343906794849929711260, 5567596199767400904172045 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois Alcover, Jan 15 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A209664-A209673.

Main diagonal of A209666 and A261718.

Sequence in context: A159313 A054910 A028562 * A180829 A227544 A094656

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

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Last modified March 30 04:42 EDT 2017. Contains 284296 sequences.