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A113953 A Jacobsthal triangle.

%I A113953 #25 Dec 04 2022 12:33:32
%S A113953 1,0,1,0,2,1,0,0,4,1,0,0,4,6,1,0,0,0,12,8,1,0,0,0,8,24,10,1,0,0,0,0,
%T A113953 32,40,12,1,0,0,0,0,16,80,60,14,1,0,0,0,0,0,80,160,84,16,1,0,0,0,0,0,
%U A113953 32,240,280,112,18,1,0,0,0,0,0,0,192,560,448,144,20,1,0,0,0,0,0,0,64,672,1120,672,180,22,1
%N A113953 A Jacobsthal triangle.
%C A113953 Rows sums are the Jacobsthal numbers A001045(n+1).
%C A113953 Antidiagonal sums are the Padovan-Jacobsthal numbers A052947.
%C A113953 Inverse is (1,xc(-2x)), c(x) the g.f. of A000108, with general term k*C(2n-k-1,n-k)(-2)^(n - k)/n.
%C A113953 Triangle read by rows given by (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 01 2013
%H A113953 D. Merlini, R. Sprugnoli, M. C. Verri, <a href="https://doi.org/10.1016/S0304-3975(98)00204-7">Strip tiling and regular grammars</a>, Theor. Comp. Sci 242 (1-2) (2000) 109-124, Table 1, p=2.
%F A113953 G.f.: 1/(1-xy(1+2x)).
%F A113953 Riordan array (1, x(1+2x)).
%F A113953 T(n,k) = 2^(n-k)*binomial(k, n-k).
%F A113953 T(n,k) = A026729(n,k)*2^(n-k). - _Philippe Deléham_, Nov 22 2006
%F A113953 T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Nov 01 2013
%e A113953 Rows begin
%e A113953   1;
%e A113953   0,  1;
%e A113953   0,  2,  1;
%e A113953   0,  0,  4,  1;
%e A113953   0,  0,  4,  6,  1;
%e A113953   0,  0,  0, 12,  8,  1;
%e A113953   0,  0,  0,  8, 24, 10,  1;
%Y A113953 A signed version is A110509.
%K A113953 easy,nonn,tabl
%O A113953 0,5
%A A113953 _Paul Barry_, Nov 09 2005