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.

1, 3, 15, 55, 216, 729, 2621, 8535, 28689, 91749, 296538, 929712, 2939063, 9093255, 28257123, 86681608, 266368959, 811501848, 2475331535, 7505567037, 22772955015, 68828023329, 208079886258, 627418618533, 1892181244828, 5696253823476, 17149663331259

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

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

G.f.: Product_{k>=1} 1 / (1 - binomial(k+2,2)*x^k). - Ilya Gutkovskiy, May 09 2021

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i)*binomial(i+2, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Mathematica

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]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

Crossrefs

Column k=3 of A261718.

Cf. A293367.

Sequence in context: A152896 A007973 A382614 * A015249 A084152 A084175

Adjacent sequences: A261734 A261735 A261736 * A261738 A261739 A261740

Keyword

nonn

Author

Alois P. Heinz, Aug 30 2015

Status

approved