A119335 - OEIS

A119335

Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

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

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OFFSET

0,25

COMMENTS

Row sums are A119336. Product of Pascal's triangle and A119337.

LINKS

Seiichi Manyama, Rows n = 0..139, flattened

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, 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

Adjacent sequences: A119332 A119333 A119334 * A119336 A119337 A119338

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 14 2006

EXTENSIONS

More terms from Seiichi Manyama, Mar 12 2019

STATUS

approved