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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.

%I A209668 #33 May 10 2021 07:41:16
%S A209668 1,1,7,55,631,8001,130453,2323483,48916087,1129559068,29442232007,
%T A209668 835245785452,26113646252773,880685234758941,32191922753658129,
%U A209668 1259701078978200555,52802268925363689079,2352843030410455053891,111343906794849929711260,5567596199767400904172045
%N 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.
%C A209668 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
%H A209668 Alois P. Heinz, <a href="/A209668/b209668.txt">Table of n, a(n) for n = 0..386</a>
%H A209668 Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%F A209668 a(n) ~ c * n^n, where c = A247551 = Product_{k>=2} 1/(1-1/k!) = 2.529477472... . - _Vaclav Kotesovec_, Nov 15 2016
%F A209668 a(n) = [x^n] Product_{k>=1} 1 / (1 - binomial(k+n-1,n-1)*x^k). - _Ilya Gutkovskiy_, May 09 2021
%p A209668 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A209668       add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
%p A209668     end:
%p A209668 a:= n-> b(n$3):
%p A209668 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 29 2015
%t A209668 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}]
%t A209668 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_ *)
%Y A209668 Cf. A209664-A209673.
%Y A209668 Main diagonal of A209666 and A261718.
%Y A209668 Cf. A261783.
%K A209668 nonn
%O A209668 0,3
%A A209668 _Wouter Meeussen_, Mar 11 2012
%E A209668 a(0)=1 prepended and a(11)-a(19) added by _Alois P. Heinz_, Aug 29 2015