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 A115990 #20 Dec 29 2023 12:56:52
%S A115990 1,1,1,3,2,1,7,5,3,1,19,13,8,4,1,51,35,22,12,5,1,141,96,61,35,17,6,1,
%T A115990 393,267,171,101,53,23,7,1,1107,750,483,291,160,77,30,8,1,3139,2123,
%U A115990 1373,839,476,244,108,38,9,1,8953,6046,3923,2423,1406,752,360,147,47,10
%N A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).
%C A115990 First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).
%C A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).
%H A115990 G. C. Greubel, <a href="/A115990/b115990.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A115990 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).
%e A115990 Triangle begins
%e A115990 1;
%e A115990 1, 1;
%e A115990 3, 2, 1;
%e A115990 7, 5, 3, 1;
%e A115990 19, 13, 8, 4, 1;
%e A115990 51, 35, 22, 12, 5, 1;
%e A115990 141, 96, 61, 35, 17, 6, 1;
%p A115990 A115990 := proc(n,k)
%p A115990 add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ;
%p A115990 end proc:
%p A115990 seq(seq(A115990(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jun 25 2023
%t A115990 Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* _G. C. Greubel_, Mar 07 2017 *)
%o A115990 (PARI) {T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ _G. C. Greubel_, May 09 2019
%o A115990 (Magma) [[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 09 2019
%o A115990 (Sage) [[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 09 2019
%o A115990 (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # _G. C. Greubel_, May 09 2019
%Y A115990 Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .
%K A115990 easy,nonn,tabl
%O A115990 0,4
%A A115990 _Paul Barry_, Feb 10 2006