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A111960 Renewal array for central trinomial numbers A002426.

%I A111960 #25 Oct 16 2022 13:44:03
%S A111960 1,1,1,3,2,1,7,7,3,1,19,20,12,4,1,51,61,40,18,5,1,141,182,135,68,25,6,
%T A111960 1,393,547,441,251,105,33,7,1,1107,1640,1428,888,420,152,42,8,1,3139,
%U A111960 4921,4572,3076,1596,654,210,52,9,1,8953,14762,14535,10456,5880,2652,966,280,63,10,1
%N A111960 Renewal array for central trinomial numbers A002426.
%C A111960 Also the convolution triangle of A002426. - _Peter Luschny_, Oct 06 2022
%F A111960 Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
%F A111960 From _Paul Barry_, May 12 2009: (Start)
%F A111960 Equals ((1-x^2)/(1+x+x^2),x/(1+x+x^2))^{-1}*(1,x/(1-x^2))=A094531*(1,x/(1-x^2)).
%F A111960 Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
%F A111960 T(n,k) = Sum_{j=0..n} C(n,j)*C((j-1)/2,(j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
%F A111960 G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)
%e A111960 Triangle T(n,k) begins:
%e A111960    1;
%e A111960    1,  1;
%e A111960    3,  2,  1;
%e A111960    7,  7,  3,  1;
%e A111960   19, 20, 12,  4, 1;
%e A111960   51, 61, 40, 18, 5, 1;
%e A111960   ...
%e A111960 From _Paul Barry_, May 12 2009: (Start)
%e A111960 Production matrix is
%e A111960   1, 1,
%e A111960   2, 1, 1,
%e A111960   0, 2, 1, 1,
%e A111960   -2, 0, 2, 1, 1,
%e A111960   0, -2, 0, 2, 1, 1,
%e A111960   4, 0, -2, 0, 2, 1, 1. (End)
%p A111960 # Uses function PMatrix from A357368. Adds a row and column above and to the left.
%p A111960 PMatrix(10, n -> A002426(n - 1)); # _Peter Luschny_, Oct 06 2022
%Y A111960 Row sums are A111961.
%Y A111960 Diagonal sums are A111962.
%Y A111960 Inverse is A111963.
%Y A111960 Factors as A007318*A111959.
%Y A111960 Column k=0 gives A002426.
%Y A111960 Cf. A026325.
%K A111960 nonn,tabl,easy
%O A111960 0,4
%A A111960 _Paul Barry_, Aug 23 2005