A293369 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 and all five letters occur at least once in the partition.

246, 4350, 44475, 369675, 2603670, 16993932, 102603315, 598010585, 3339393990, 18294499370, 97818690363, 517148440820, 2694756962105, 13947673300505, 71555207694490, 365571598248050, 1857609632705200, 9414446265923035, 47553294423090160, 239799029393392505

Offset: 5

Author

Alois P. Heinz, Oct 07 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=5 of A261719.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
seq(a(n), n=5..30);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k] Binomial[i + k - 1, k - 1]]]];
a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
a /@ Range[5, 30] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 5^n, where c = 4.1548340497015786311470026968208254860294132084317763408428889184148319... - Vaclav Kotesovec, Oct 11 2017