login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A119335
Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).
8
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 11, 17, 11, 1, 1, 1, 1, 1, 1, 21, 41, 41, 21, 1, 1, 1, 1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1, 1, 1, 1, 57, 141, 201, 201, 141, 57, 1, 1, 1
OFFSET
0,25
COMMENTS
Row sums are A119336. Product of Pascal's triangle and A119337.
LINKS
FORMULA
Column k has g.f. (x^k/(1-x)) * Sum_{j=0..k} C(k,3j)(x/(1-x))^(3j).
EXAMPLE
Triangle begins
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1;
1, 1, 1, 2, 1, 1, 1;
1, 1, 1, 5, 5, 1, 1, 1;
1, 1, 1, 11, 17, 11, 1, 1, 1;
1, 1, 1, 21, 41, 41, 21, 1, 1, 1;
1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1;
MATHEMATICA
T[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023 *)
CROSSREFS
T(2n,n) gives A119363.
Cf. A119326.
Sequence in context: A225372 A184879 A373201 * A155869 A176564 A237717
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, May 14 2006
EXTENSIONS
More terms from Seiichi Manyama, Mar 12 2019
STATUS
approved