This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A108756 #52 Sep 08 2022 08:45:19
%S A108756 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,
%T A108756 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,
%U A108756 120,165,78,91,16,17,1,1,6,21,126,210,220,286,105,120,18,19,1,1
%N A108756 A triangle related to the Jacobsthal polynomials.
%C A108756 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.
%H A108756 G. C. Greubel, <a href="/A108756/b108756.txt">Rows n = 0..100 of triangle, flattened</a>
%H A108756 D. Stutson, V. Kocic, and G. Arora, <a href="/A108756/a108756.pdf">A Few Identities involving Jacobsthal polynomials</a>, Xavier University of Louisiana, preprint, 2005.
%F A108756 Number triangle: T(n, k) = binomial(floor((n + k + 1)/2) + k, floor((n + k)/2 - k)) for 0 <= k <= n.
%F A108756 From _Petros Hadjicostas_, May 30 2019: (Start)
%F A108756 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.)
%F A108756 G.f. for column k >= 0: (1 + x - x^2) * x^k/(1 - x^2)^(2*k + 2).
%F A108756 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.
%F A108756 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.)
%F A108756 (End)
%e A108756 Triangle begins (with rows n >= 0 and columns k >= 0) as follows:
%e A108756 1;
%e A108756 1, 1;
%e A108756 1, 1, 1;
%e A108756 2, 3, 1, 1;
%e A108756 1, 4, 5, 1, 1;
%e A108756 3, 6, 6, 7, 1, 1;
%e A108756 1, 10, 15, 8, 9, 1, 1;
%e A108756 4, 10, 21, 28, 10, 11, 1, 1;
%e A108756 1, 20, 35, 36, 45, 12, 13, 1, 1;
%e A108756 5, 15, 56, 84, 55, 66, 14, 15, 1, 1; ...
%t A108756 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 *)
%o A108756 (PARI) {T(n,k) = binomial(floor((n+k+1)/2)+k, floor((n+k)/2)-k)}; \\ _G. C. Greubel_, May 29 2019
%o A108756 (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
%o A108756 (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
%Y A108756 Cf. A000045, A102426, A108742.
%K A108756 easy,nonn,tabl
%O A108756 0,7
%A A108756 _Paul Barry_, Jun 22 2005
%E A108756 More terms from _Petros Hadjicostas_, May 29 2019