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 A091597 #20 Sep 08 2022 08:45:13
%S A091597 1,1,1,3,2,1,5,5,3,1,11,10,8,4,1,21,21,18,12,5,1,43,42,39,30,17,6,1,
%T A091597 85,85,81,69,47,23,7,1,171,170,166,150,116,70,30,8,1,341,341,336,316,
%U A091597 266,186,100,38,9,1,683,682,677,652,582,452,286,138,47,10,1
%N A091597 Triangle read by rows: T(n,0) = A001045(n+1), T(n,n)=1, T(n,m) = T(n-1,m-1) + T(n-1,m).
%C A091597 A Jacobsthal-Pascal triangle.
%C A091597 Equals triangle M * Pascal's triangle, M = an infinite lower triangular Toeplitz matrix with A078008: [1, 0, 2, 2, 6, 10, 22, 42, ...] in every column. - _Gary W. Adamson_, May 25 2009
%H A091597 G. C. Greubel, <a href="/A091597/b091597.txt">Rows n = 0..100 of triangle, flattened</a>
%F A091597 Number triangle: T(n, k) = Sum_{j=0..n} binomial(n-j, k+j)2^j.
%F A091597 Riordan array: (1/(1-x-2*x^2), x/(1-x)).
%F A091597 k-th column has g.f. (1/(1-x-2*x^2))*(x/(1-x))^k.
%F A091597 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - 2*T(n-3,k) - 2*T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=T(2,2)=1, T(2,0)=3, T(2,1)=2, T(n,k)=0 if k < 0 or if k > n. - _Philippe Deléham_, Jan 11 2014
%e A091597 Triangle begins as:
%e A091597 1;
%e A091597 1, 1;
%e A091597 3, 2, 1;
%e A091597 5, 5, 3, 1;
%e A091597 11, 10, 8, 4, 1;
%e A091597 21, 21, 18, 12, 5, 1;
%e A091597 43, 42, 39, 30, 17, 6, 1;
%e A091597 85, 85, 81, 69, 47, 23, 7, 1;
%e A091597 171, 170, 166, 150, 116, 70, 30, 8, 1;
%e A091597 341, 341, 336, 316, 266, 186, 100, 38, 9, 1;
%p A091597 A091597 := proc(n,k)
%p A091597 if k = 0 then
%p A091597 A001045(n+1) ;
%p A091597 elif k = n then
%p A091597 1 ;
%p A091597 elif k <0 or k > n then
%p A091597 0 ;
%p A091597 else
%p A091597 procname(n-1,k-1)+procname(n-1,k) ;
%p A091597 end if;
%p A091597 end proc: # _R. J. Mathar_, Oct 05 2012
%t A091597 Table[Sum[Binomial[n-j, k+j]*2^j, {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 04 2019 *)
%o A091597 (PARI) {T(n,k) = sum(j=0, n, 2^j*binomial(n-j, k+j))}; \\ _G. C. Greubel_, Jun 04 2019
%o A091597 (Magma) [[(&+[2^j*Binomial(n-j, k+j): j in [0..n]]): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Jun 04 2019
%o A091597 (Sage) [[sum(2^j*binomial(n-j, k+j) for j in (0..n)) for k in (0..n)] for n in [0..12]] # _G. C. Greubel_, Jun 04 2019
%o A091597 (GAP) Flat(List([0..12], n->List([0..n], k-> Sum([0..n], j-> 2^j*Binomial(n-j, k+j)) ))); # _G. C. Greubel_, Jun 04 2019
%Y A091597 Columns include A001045, A000975, A011377.
%Y A091597 Row sums are A059570.
%Y A091597 Cf. A078008. - _Gary W. Adamson_, May 25 2009
%K A091597 easy,nonn,tabl
%O A091597 0,4
%A A091597 _Paul Barry_, Jan 23 2004