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
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.
Programs
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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
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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
Comments