|
| |
|
|
A191239
|
|
Triangle T(n,k) = coefficient of x^n in expansion of (x+x^2+2*x^3)^k.
|
|
0
|
|
|
|
1, 1, 1, 2, 2, 1, 0, 5, 3, 1, 0, 4, 9, 4, 1, 0, 4, 13, 14, 5, 1, 0, 0, 18, 28, 20, 6, 1, 0, 0, 12, 49, 50, 27, 7, 1, 0, 0, 8, 56, 105, 80, 35, 8, 1, 0, 0, 0, 56, 161, 195, 119, 44, 9, 1, 0, 0, 0, 32, 210, 366, 329, 168, 54, 10, 1, 0, 0, 0, 16, 200, 581, 721, 518, 228, 65, 11, 1, 0, 0, 0, 0, 160, 732, 1337, 1288, 774, 300, 77, 12, 1, 0, 0, 0, 0, 80, 780, 2045, 2716, 2142, 1110, 385, 90, 13, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
1. Riordan Array (1,x+x^2+2*x^3) without first column.
2. Riordan Array (1+x+2*x^3,x+x^2+2*x^3) numbering triangle (0,0).
3. Bell Polynomial of second kind B(n,k){1,2,12,0,0,0,...,0}=n!/k!*T(n,k).
4. For the g.f. 1/(1-x-x^2-2*x^3) we have a(n)=sum(k=1..n, T(n,k)) (see A077947)
For more formulas see preprints.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = Sum_{j=0..k} binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j).
|
|
|
EXAMPLE
|
Triangle begins:
1,
1,1,
2,2,1,
0,5,3,1,
0,4,9,4,1,
0,4,13,14,5,1,
0,0,18,28,20,6,1,
|
|
|
PROG
|
(Maxima)
T(n, k):=sum(binomial(j, n-3*k+2*j)*2^(k-j)*binomial(k, j), j, 0, k);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
