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A261739 Number of partitions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order.

%I A261739 #14 May 10 2021 04:49:08
%S A261739 1,5,40,235,1470,8001,45865,241870,1307055,6783210,35510502,181665635,
%T A261739 934801705,4741017595,24118500815,121693135003,614889556920,
%U A261739 3091596201560,15557885702390,78054925105630,391798489621630,1963104427709830,9838685572501515
%N A261739 Number of partitions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order.
%H A261739 Alois P. Heinz, <a href="/A261739/b261739.txt">Table of n, a(n) for n = 0..1000</a>
%F A261739 a(n) ~ c * 5^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)*(k+3)*(k+4)/(24*5^k)) = 4.1548340497015786311470026968208254860294132084317763408428889184148319... - _Vaclav Kotesovec_, Oct 11 2017, updated May 10 2021
%F A261739 G.f.: Product_{k>=1} 1 / (1 - binomial(k+4,4)*x^k). - _Ilya Gutkovskiy_, May 09 2021
%p A261739 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261739       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+4, 4))))
%p A261739     end:
%p A261739 a:= n-> b(n$2):
%p A261739 seq(a(n), n=0..30);
%t A261739 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]*Binomial[i + 4, 4]]]];
%t A261739 a[n_] := b[n, n];
%t A261739 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A261739 Column k=5 of A261718.
%K A261739 nonn
%O A261739 0,2
%A A261739 _Alois P. Heinz_, Aug 30 2015