A226875 - OEIS

%I #23 Sep 21 2017 11:30:10

%S 1,1,3,10,47,246,882,3921,18223,84790,432518,1863951,8892842,42656147,

%T 204204353,1025014815,4728033983,22948258742,111605089014,

%U 541696830843,2708218059022,12861557284425,62938669549583,308273057334413,1508708926286914,7533652902408071

%N Number of n-length words w over a 5-ary alphabet {a1,a2,...,a5} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a5) >= 0, where #(w,x) counts the letters x in word w.

%H Alois P. Heinz, <a href="/A226875/b226875.txt">Table of n, a(n) for n = 0..1000</a>

%F Conjecture: a(n) ~ 5^n/5!. - _Vaclav Kotesovec_, Mar 07 2014

%p b:= proc(n, i, t) option remember;

%p       `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))

%p     end:

%p a:= n-> n!*b(n, 0, 5):

%p seq(a(n), n=0..30);

%t Table[Sum[Sum[Sum[Sum[Sum[If[i+j+k+l+m==n,n!/i!/j!/k!/l!/m!,0],{m,0,l}],{l,0,k}],{k,0,j}],{j,0,i}],{i,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 01 2013 *)

%t CoefficientList[Series[(HypergeometricPFQ[{},{},x]^5 + 10*HypergeometricPFQ[{},{},x]^3*HypergeometricPFQ[{},{1},x^2] + 20*HypergeometricPFQ[{},{},x]^2*HypergeometricPFQ[{},{1,1},x^3] + 20*HypergeometricPFQ[{},{1},x^2]*HypergeometricPFQ[{},{1,1},x^3] + 15*HypergeometricPFQ[{},{1},x^2]^2*HypergeometricPFQ[{},{},x] + 30*HypergeometricPFQ[{},{1,1,1},x^4]*HypergeometricPFQ[{},{},x] + 24*HypergeometricPFQ[{},{1,1,1,1},x^5])/5!,{x,0,20}],x]*Range[0,20]! (* more efficient, _Vaclav Kotesovec_, Jul 01 2013 *)

%Y Column k=5 of A226873.

%Y Cf. A027306, A092255, A092429.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jun 21 2013