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A258498 Number of words of length 2n such that the index set of occurring letters is {1, 2, ..., k}, all letters are introduced in ascending order, and the words can be built by repeatedly inserting doublets into the initially empty word. 4
1, 1, 3, 15, 105, 933, 9988, 124449, 1761287, 27813479, 483482018, 9153385959, 187129080977, 4102129113670, 95861136747795, 2376234441556411, 62216635372018209, 1714347701138957189, 49553280367466054768, 1498300016807379304877, 47270249397381096576643 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..447

FORMULA

a(n) = Sum_{k=0..n} A256117(n,k).

a(n) ~ Bell(n-1)*Catalan(n) ~ n^n * exp(n/LambertW(n)-1-n) * 4^n / (sqrt(Pi) * sqrt(1+LambertW(n)) * LambertW(n)^(n-1) * n^(5/2)). - Vaclav Kotesovec, Jun 02 2015

EXAMPLE

a(3) = 15: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba, aabbcc, aabccb, abbacc, abbcca, abccba.

MAPLE

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

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

    end:

T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):

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

seq(a(n), n=0..25);

MATHEMATICA

A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];

T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];

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

a /@ Range[0, 25] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

CROSSREFS

Row sums of A256117.

Cf. A294603, A321031.

Sequence in context: A291744 A246860 A249014 * A189919 A251598 A338725

Adjacent sequences:  A258495 A258496 A258497 * A258499 A258500 A258501

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 31 2015

STATUS

approved

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)