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A130456 Triangle, read by rows of 2n+1 terms, where row n+1 is generated by taking partial sums of row n and then append a zero followed by the final term in the partial sums of row n, for n>=0, with T(0,0)=1.

%I A130456 #5 Jun 27 2016 07:53:57
%S A130456 1,1,0,1,1,1,2,0,2,1,2,4,4,6,0,6,1,3,7,11,17,17,23,0,23,1,4,11,22,39,
%T A130456 56,79,79,102,0,102,1,5,16,38,77,133,212,291,393,393,495,0,495,1,6,22,
%U A130456 60,137,270,482,773,1166,1559,2054,2054,2549,0,2549,1,7,29,89,226,496,978
%N A130456 Triangle, read by rows of 2n+1 terms, where row n+1 is generated by taking partial sums of row n and then append a zero followed by the final term in the partial sums of row n, for n>=0, with T(0,0)=1.
%C A130456 Final term in row n equals A098746(n) = number of permutations of [1..n] which avoid 4231 and 42513.
%e A130456 Triangle begins:
%e A130456 1;
%e A130456 1, 0, 1;
%e A130456 1, 1, 2, 0, 2;
%e A130456 1, 2, 4, 4, 6, 0, 6;
%e A130456 1, 3, 7, 11, 17, 17, 23, 0, 23;
%e A130456 1, 4, 11, 22, 39, 56, 79, 79, 102, 0, 102;
%e A130456 1, 5, 16, 38, 77, 133, 212, 291, 393, 393, 495, 0, 495;
%e A130456 1, 6, 22, 60, 137, 270, 482, 773, 1166, 1559, 2054, 2054, 2549, 0, 2549; ...
%o A130456 (PARI) {T(n,k)=local(A=[1],B);if(n==0,if(k==0,1,0),for(j=1,n, B=Vec(Ser(A)/(1-x));A=concat(concat(B,0),B[ #B]));A[k+1])}
%Y A130456 Cf. A098746 (final term in rows).
%K A130456 nonn,tabf
%O A130456 0,7
%A A130456 _Paul D. Hanna_, May 26 2007