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A261743 Number of partitions of n where each part i is marked with a word of length i over a nonary alphabet whose letters appear in alphabetical order.

%I A261743 #11 May 10 2021 06:24:41
%S A261743 1,9,126,1299,14211,136611,1373127,12838293,122478147,1129559068,
%T A261743 10495764324,95773104459,877873080195,7963150929030,72400207009635,
%U A261743 654588661768353,5924851016703093,53460853371243261,482688774419853026,4350478100196378069,39224153751141474936
%N A261743 Number of partitions of n where each part i is marked with a word of length i over a nonary alphabet whose letters appear in alphabetical order.
%H A261743 Alois P. Heinz, <a href="/A261743/b261743.txt">Table of n, a(n) for n = 0..1000</a>
%F A261743 a(n) ~ c * 9^n, where c = Product_{k>=2} 1/(1 - binomial(k+8,8)/9^k) = 3.23950351986835655716873222462341048089067679826... - _Vaclav Kotesovec_, Oct 11 2017, updated May 10 2021
%F A261743 G.f.: Product_{k>=1} 1 / (1 - binomial(k+8,8)*x^k). - _Ilya Gutkovskiy_, May 10 2021
%p A261743 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261743       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+8, 8))))
%p A261743     end:
%p A261743 a:= n-> b(n$2):
%p A261743 seq(a(n), n=0..30);
%Y A261743 Column k=9 of A261718.
%K A261743 nonn
%O A261743 0,2
%A A261743 _Alois P. Heinz_, Aug 30 2015