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A261737 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order.

%I A261737 #20 Nov 16 2024 02:02:20
%S A261737 1,3,15,55,216,729,2621,8535,28689,91749,296538,929712,2939063,
%T A261737 9093255,28257123,86681608,266368959,811501848,2475331535,7505567037,
%U A261737 22772955015,68828023329,208079886258,627418618533,1892181244828,5696253823476,17149663331259
%N A261737 Number of partitions of n where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order.
%H A261737 Alois P. Heinz, <a href="/A261737/b261737.txt">Table of n, a(n) for n = 0..1000</a>
%F A261737 a(n) ~ c * 3^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)/(2*3^k)) = 6.84620607349852135789816336867607014231681538613599316638081993041973716978... . - _Vaclav Kotesovec_, Nov 15 2016, updated May 10 2021
%F A261737 G.f.: Product_{k>=1} 1 / (1 - binomial(k+2,2)*x^k). - _Ilya Gutkovskiy_, May 09 2021
%p A261737 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261737       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+2, 2))))
%p A261737     end:
%p A261737 a:= n-> b(n$2):
%p A261737 seq(a(n), n=0..30);
%t A261737 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 + 2, 2]]]];
%t A261737 a[n_] := b[n, n];
%t A261737 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y A261737 Column k=3 of A261718.
%Y A261737 Cf. A293367.
%K A261737 nonn
%O A261737 0,2
%A A261737 _Alois P. Heinz_, Aug 30 2015