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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
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
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}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz, updated Jan 01 2021 *)
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.
Sequence in context: A021836 A255306 A072551 * A352372 A219034 A256116
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 15 2015
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)