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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.

%I A293369 #11 Dec 12 2020 04:44:33
%S A293369 246,4350,44475,369675,2603670,16993932,102603315,598010585,
%T A293369 3339393990,18294499370,97818690363,517148440820,2694756962105,
%U A293369 13947673300505,71555207694490,365571598248050,1857609632705200,9414446265923035,47553294423090160,239799029393392505
%N 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.
%H A293369 Alois P. Heinz, <a href="/A293369/b293369.txt">Table of n, a(n) for n = 5..1000</a>
%F A293369 a(n) ~ c * 5^n, where c = 4.1548340497015786311470026968208254860294132084317763408428889184148319... - _Vaclav Kotesovec_, Oct 11 2017
%p A293369 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A293369       b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
%p A293369     end:
%p A293369 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(5):
%p A293369 seq(a(n), n=5..30);
%t A293369 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]]]];
%t A293369 a[n_] := With[{k = 5}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A293369 a /@ Range[5, 30] (* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y A293369 Column k=5 of A261719.
%K A293369 nonn
%O A293369 5,1
%A A293369 _Alois P. Heinz_, Oct 07 2017