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A261738 Number of partitions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order.

%I A261738 #14 May 10 2021 04:45:01
%S A261738 1,4,26,124,631,2780,12954,55196,241634,1012196,4280046,17636252,
%T A261738 73157709,298342936,1220952044,4947485904,20079338277,80987461760,
%U A261738 326986050564,1314939934216,5290893771329,21236552526364,85263892578686,341801704446572,1370448001291679
%N A261738 Number of partitions of n where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order.
%H A261738 Alois P. Heinz, <a href="/A261738/b261738.txt">Table of n, a(n) for n = 0..1000</a>
%F A261738 a(n) ~ c * 4^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)*(k+3)/(3*2^(2*k+1))) = 4.90673361196637084263021203165784685586076564592828337755053385514766785... - _Vaclav Kotesovec_, Oct 11 2017, updated May 10 2021
%F A261738 G.f.: Product_{k>=1} 1 / (1 - binomial(k+3,3)*x^k). - _Ilya Gutkovskiy_, May 09 2021
%p A261738 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261738       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+3, 3))))
%p A261738     end:
%p A261738 a:= n-> b(n$2):
%p A261738 seq(a(n), n=0..30);
%t A261738 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 + 3, 3]]]];
%t A261738 a[n_] := b[n, n];
%t A261738 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A261738 Column k=4 of A261718.
%K A261738 nonn
%O A261738 0,2
%A A261738 _Alois P. Heinz_, Aug 30 2015