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A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).

%I A091598 #10 Jul 22 2019 06:27:54
%S A091598 1,0,1,2,1,1,2,3,2,1,6,5,5,3,1,10,11,10,8,4,1,22,21,21,18,12,5,1,42,
%T A091598 43,42,39,30,17,6,1,86,85,85,81,69,47,23,7,1,170,171,170,166,150,116,
%U A091598 70,30,8,1,342,341,341,336,316,266,186,100,38,9,1,682,683,682,677,652,582,452,286,138,47,10,1
%N A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).
%C A091598 A Jacobsthal-Pascal triangle.
%H A091598 G. C. Greubel, <a href="/A091598/b091598.txt">Rows n = 0..20 of triangle, flattened</a>
%F A091598 k-th column has e.g.f. ((1-x)/(1-x-x^2))*(x/(1-x))^k.
%e A091598 Triangle starts as:
%e A091598    1;
%e A091598    0,  1;
%e A091598    2,  1,  1;
%e A091598    2,  3,  2,  1;
%e A091598    6,  5,  5,  3,  1;
%e A091598   10, 11, 10,  8,  4,  1;
%e A091598   22, 21, 21, 18, 12,  5, 1;
%e A091598   42, 43, 42, 39, 30, 17, 6, 1; ...
%t A091598 T[n_, k_]:= If[k==0, (2^n + 2*(-1)^n)/3, If[k<0 || k>n, 0, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 04 2019 *)
%o A091598 (PARI) {T(n,k) = if(k==0, (2^n + 2*(-1)^n)/3, if(k<0 || k>n, 0, T(n-1,k-1) + T(n-1,k)))}; \\ _G. C. Greubel_, Jun 04 2019
%o A091598 (Sage)
%o A091598 def T(n, k):
%o A091598     if (k<0 or k>n): return 0
%o A091598     elif (k==0): return (2^n + 2*(-1)^n)/3
%o A091598     else: return T(n-1, k-1) + T(n-1, k)
%o A091598 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jun 04 2019
%Y A091598 Columns include A078008, A001045, A000975, A011377. Row sums give A084219.
%Y A091598 Cf. A091597.
%K A091598 easy,nonn,tabl
%O A091598 0,4
%A A091598 _Paul Barry_, Jan 23 2004