A108756 A triangle related to the Jacobsthal polynomials.

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 4, 5, 1, 1, 3, 6, 6, 7, 1, 1, 1, 10, 15, 8, 9, 1, 1, 4, 10, 21, 28, 10, 11, 1, 1, 1, 20, 35, 36, 45, 12, 13, 1, 1, 5, 15, 56, 84, 55, 66, 14, 15, 1, 1, 1, 35, 70, 120, 165, 78, 91, 16, 17, 1, 1, 6, 21, 126, 210, 220, 286, 105, 120, 18, 19, 1, 1

Offset

0,7

Comments

Riordan array ((1 + x - x^2)/(1 - x^2)^2, x/(1 - x^2)^2). Row sums are A108742. Diagonal sums are Fibonacci(n+1) = A000045(n+1). Corresponding diagonals triangle is A102426.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

D. Stutson, V. Kocic, and G. Arora, A Few Identities involving Jacobsthal polynomials, Xavier University of Louisiana, preprint, 2005.

Formula

Number triangle: T(n, k) = binomial(floor((n + k + 1)/2) + k, floor((n + k)/2 - k)) for 0 <= k <= n.

From Petros Hadjicostas, May 30 2019: (Start)

Bivariate g.f.: Sum_{n, k > = 0} T(n,k) * x^n * y^k = (1 + x - x^2)/((1 - x^2)^2 - x * y). (Here, we assume T(n, k) = 0 for n < k. Because T(n, k) = A102426(n + 1 + k, k), we may use Tom Copeland's g.f. of the latter array, to get the g.f. of the current triangular array.)

G.f. for column k >= 0: (1 + x - x^2) * x^k/(1 - x^2)^(2*k + 2).

Recurrence: T(n, k) = T(n - 2, k) + T(n - (1 - (-1)^(n + k))/2, k - (1 + (-1)^(n + k))/2), for n >= 3 and 1 <= k <= n - 2, starting with T(n, n) = 1 = T(n + 1, n) for n >= 0, T(n, 0) = 1 when n is even >= 0, and T(n, 0) = (n + 1)/2 when n is odd >= 1.

Another recurrence: T(n, k) = T(n - 1, k - 1) + 2*T(n - 2, k) - T(n - 4, k) for n >= 4 and 1 <= k <= n - 4. (This follows from the fact that the denominator of the bivariate g.f. is x^0 * y^0 - x^1 * y^1 - 2 * x^2 * y^0 - x^4 * y^0.)

(End)

Example

Triangle begins (with rows n >= 0 and columns k >= 0) as follows:
  1;
  1,  1;
  1,  1,  1;
  2,  3,  1,  1;
  1,  4,  5,  1,  1;
  3,  6,  6,  7,  1,  1;
  1, 10, 15,  8,  9,  1,  1;
  4, 10, 21, 28, 10, 11,  1,  1;
  1, 20, 35, 36, 45, 12, 13,  1, 1;
  5, 15, 56, 84, 55, 66, 14, 15, 1, 1; ...

Mathematica

Table[Binomial[Floor[(n+k+1)/2]+k, Floor[(n+k)/2]-k], {n, 0, 12}, {k, 0, n} ]//Flatten (* G. C. Greubel, May 29 2019 *)

Prog

(PARI) {T(n, k) = binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k)}; \\ G. C. Greubel, May 29 2019
(Magma) [[Binomial(Floor((n+k+1)/2)+k, Floor((n+k)/2)-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 29 2019
(Sage) [[binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 29 2019

Crossrefs

Cf. A000045, A102426, A108742.

Sequence in context: A261019 A140737 A354009 * A106178 A305807 A366292

Adjacent sequences: A108753 A108754 A108755 * A108757 A108758 A108759

Keyword

easy,nonn,tabl

Author

Paul Barry, Jun 22 2005

Extensions

More terms from Petros Hadjicostas, May 29 2019

Status

approved