A261739 Number of partitions of n where each part i is marked with a word of length i over a quinary alphabet whose letters appear in alphabetical order.

1, 5, 40, 235, 1470, 8001, 45865, 241870, 1307055, 6783210, 35510502, 181665635, 934801705, 4741017595, 24118500815, 121693135003, 614889556920, 3091596201560, 15557885702390, 78054925105630, 391798489621630, 1963104427709830, 9838685572501515

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=5 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+4, 4))))
    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 + 4, 4]]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 5^n, where c = Product_{k>=2} 1/(1 - (k+1)*(k+2)*(k+3)*(k+4)/(24*5^k)) = 4.1548340497015786311470026968208254860294132084317763408428889184148319... - Vaclav Kotesovec, Oct 11 2017, updated May 10 2021

G.f.: Product_{k>=1} 1 / (1 - binomial(k+4,4)*x^k). - Ilya Gutkovskiy, May 09 2021