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 A115991 #16 Jun 25 2023 13:28:42
%S A115991 1,1,1,5,3,1,13,9,5,1,49,31,17,7,1,161,105,61,29,9,1,581,371,217,111,
%T A115991 45,11,1,2045,1313,781,417,189,65,13,1,7393,4719,2825,1551,753,303,89,
%U A115991 15,1,26689,17041,10277,5757,2921,1289,461,117,17,1
%N A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j)*2^(n-j).
%C A115991 First column is A084601 with e.g.f. exp(x) Bessel_I(0,2*sqrt(2)x). Row sums are A098518(n+1) with e.g.f. dif(exp(x) Bessel_I(1,2*sqrt(2)x)/sqrt(2)).
%C A115991 Riordan array (1/sqrt(1-2*x-7*x^2), (1+x-sqrt(1-2*x-7*x^2))/2).
%H A115991 G. C. Greubel, <a href="/A115991/b115991.txt">Rows n = 0..100 of triangle, flattened</a>
%e A115991 Triangle begins as:
%e A115991 1;
%e A115991 1, 1;
%e A115991 5, 3, 1;
%e A115991 13, 9, 5, 1;
%e A115991 49, 31, 17, 7, 1;
%e A115991 161, 105, 61, 29, 9, 1;
%e A115991 581, 371, 217, 111, 45, 11, 1;
%p A115991 A115991 := proc(n,k)
%p A115991 add(binomial(n-k,j-k)*binomial(j,n-j)*2^(n-j),j=0..n) ;
%p A115991 end proc:
%p A115991 seq(seq(A115991(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jun 25 2023
%t A115991 Table[Sum[Binomial[n-k, j-k]*Binomial[j, n-j]*2^(n-j), {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 09 2019 *)
%o A115991 (PARI) {T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j))}; \\ _G. C. Greubel_, May 09 2019
%o A115991 (Magma) [[(&+[Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 09 2019
%o A115991 (Sage) [[sum(binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 09 2019
%o A115991 (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j)) ))); # _G. C. Greubel_, May 09 2019
%Y A115991 Cf. A084601 (k=0), A098518.
%K A115991 easy,nonn,tabl
%O A115991 0,4
%A A115991 _Paul Barry_, Feb 10 2006