login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A256117 Number T(n,k) of length 2n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
1, 0, 1, 0, 1, 2, 0, 1, 9, 5, 0, 1, 34, 56, 14, 0, 1, 125, 465, 300, 42, 0, 1, 461, 3509, 4400, 1485, 132, 0, 1, 1715, 25571, 55692, 34034, 7007, 429, 0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430, 0, 1, 24309, 1325609, 7488228, 11187462, 6191808, 1447992, 143208, 4862 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

In general, column k>2 is asymptotic to (4*(k-1))^n / ((k-2)^2 * (k-2)! * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(n,k) = Sum_{i=0..k} (-1)^i * A183135(n,k-i) / (i!*(k-i)!).

T(n,k) = A256116(n,k) / (k-1)! for k > 0.

EXAMPLE

T(0,0) = 1: (the empty word).

T(1,1) = 1: aa.

T(2,1) = 1: aaaa.

T(2,2) = 2: aabb, abba.

T(3,1) = 1: aaaaaa.

T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.

T(3,3) = 5: aabbcc, aabccb, abbacc, abbcca, abccba.

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1,    2;

  0, 1,    9,      5;

  0, 1,   34,     56,     14;

  0, 1,  125,    465,    300,     42;

  0, 1,  461,   3509,   4400,   1485,    132;

  0, 1, 1715,  25571,  55692,  34034,   7007,   429;

  0, 1, 6434, 184232, 657370, 647920, 231868, 32032, 1430;

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):

seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

Unprotect[Power]; 0^0 = 1; A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[ Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]]; T[n_, k_] := Sum[ (-1)^i* A[n, k-i]/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Feb 22 2016, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A057427, A010763(n-1) (for n>1), A258490, A258491, A258492, A258493, A258494, A258495, A258496, A258497.

Main diagonal gives A000108.

T(n+2,n+1) gives A002055(n+5).

Row sums give A258498.

T(2n,n) gives A258499.

Cf. A183135, A256116, A256311.

Sequence in context: A021836 A255306 A072551 * A219034 A256116 A185410

Adjacent sequences:  A256114 A256115 A256116 * A256118 A256119 A256120

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 15 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)