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

Author

Paul Barry, Jun 22 2005

Keywords

• easy
• nonn
• tabl

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.

Examples

			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; ...
		

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.

Crossrefs

Cf. A000045, A102426, A108742.

Programs

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

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 *)

PARI

{T(n,k) = binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k)}; \\ 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

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)

Extensions

More terms from Petros Hadjicostas, May 29 2019