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.

1, 4, 26, 124, 631, 2780, 12954, 55196, 241634, 1012196, 4280046, 17636252, 73157709, 298342936, 1220952044, 4947485904, 20079338277, 80987461760, 326986050564, 1314939934216, 5290893771329, 21236552526364, 85263892578686, 341801704446572, 1370448001291679

Offset: 0

Author

Alois P. Heinz, Aug 30 2015

Keywords

• nonn

Links

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

Crossrefs

Column k=4 of A261718.

Programs

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+3, 3))))
    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 + 3, 3]]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

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

G.f.: Product_{k>=1} 1 / (1 - binomial(k+3,3)*x^k). - Ilya Gutkovskiy, May 09 2021