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A261740 Number of partitions of n where each part i is marked with a word of length i over a senary alphabet whose letters appear in alphabetical order.

%I A261740 #11 May 10 2021 04:54:20
%S A261740 1,6,57,398,2955,19158,130453,820554,5280204,32711022,204324819,
%T A261740 1249546656,7682267669,46625705988,283766862009,1714704081724,
%U A261740 10374896682273,62511439251768,376943252871343,2267304042230202,13643684237963994,81983795625450144
%N A261740 Number of partitions of n where each part i is marked with a word of length i over a senary alphabet whose letters appear in alphabetical order.
%H A261740 Alois P. Heinz, <a href="/A261740/b261740.txt">Table of n, a(n) for n = 0..1000</a>
%F A261740 a(n) ~ c * 6^n, where c = Product_{k>=2} 1/(1 - binomial(k+5,5)/6^k) = 3.760725122262068858184072984846959348360490081749654779894152320389687335... - _Vaclav Kotesovec_, Oct 11 2017, updated May 10 2021
%F A261740 G.f.: Product_{k>=1} 1 / (1 - binomial(k+5,5)*x^k). - _Ilya Gutkovskiy_, May 09 2021
%p A261740 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261740       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+5, 5))))
%p A261740     end:
%p A261740 a:= n-> b(n$2):
%p A261740 seq(a(n), n=0..30);
%Y A261740 Column k=6 of A261718.
%K A261740 nonn
%O A261740 0,2
%A A261740 _Alois P. Heinz_, Aug 30 2015