A294603 - OEIS

%I #14 Mar 19 2022 06:37:22

%S 1,1,3,20,231,3864,85360,2353546,77963599,3019479344,133966276692,

%T 6702399275538,373406941221160,22930441709648290,1539004344848618466,

%U 112089683771614695478,8805334896381292460191,742162775145283382779168,66809386370870410069346476

%N Number of words of semilength n over n-ary alphabet, either empty or beginning with the first letter of the alphabet, such that the index set of occurring letters is an integer interval [1, k], that can be built by repeatedly inserting doublets into the initially empty word.

%H Alois P. Heinz, <a href="/A294603/b294603.txt">Table of n, a(n) for n = 0..350</a>

%F a(n) = Sum_{k=0..n} A256116(n,k).

%e a(0) = 1: the empty word.

%e a(1) = 1: aa.

%e a(2) = 3: aaaa, aabb, abba.

%e a(3) = 20: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, aabbcc, aabccb, aacbbc, aaccbb, abaaba, abbaaa, abbabb, abbacc, abbbba, abbcca, abccba, acbbca, accabb, accbba.

%p A:= proc(n, k) option remember; `if`(n=0, 1, k/n*

%p       add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))

%p     end:

%p T:= proc(n, k) option remember;

%p       add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/`if`(k=0, 1, k)

%p     end:

%p a:= n-> add(T(n, k), k=0..n):

%p seq(a(n), n=0..20);

%t A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*

%t      Sum[Binomial[2*n, j]*(n-j) *If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];

%t T[n_, k_] := T[n, k] =

%t      Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k == 0, 1, k];

%t a[n_] := Sum[T[n, k], {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 19 2022, after _Alois P. Heinz_ *)

%Y Row sums of A256116.

%Y Cf. A258498.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 03 2017