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Renewal array for central trinomial numbers A002426.
4

%I #25 Oct 16 2022 13:44:03

%S 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 1,393,547,441,251,105,33,7,1,1107,1640,1428,888,420,152,42,8,1,3139,

%U 4921,4572,3076,1596,654,210,52,9,1,8953,14762,14535,10456,5880,2652,966,280,63,10,1

%N Renewal array for central trinomial numbers A002426.

%C Also the convolution triangle of A002426. - _Peter Luschny_, Oct 06 2022

%F Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).

%F From _Paul Barry_, May 12 2009: (Start)

%F 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 Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));

%F 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 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 Triangle T(n,k) begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 7, 7, 3, 1;

%e 19, 20, 12, 4, 1;

%e 51, 61, 40, 18, 5, 1;

%e ...

%e From _Paul Barry_, May 12 2009: (Start)

%e Production matrix is

%e 1, 1,

%e 2, 1, 1,

%e 0, 2, 1, 1,

%e -2, 0, 2, 1, 1,

%e 0, -2, 0, 2, 1, 1,

%e 4, 0, -2, 0, 2, 1, 1. (End)

%p # Uses function PMatrix from A357368. Adds a row and column above and to the left.

%p PMatrix(10, n -> A002426(n - 1)); # _Peter Luschny_, Oct 06 2022

%Y Row sums are A111961.

%Y Diagonal sums are A111962.

%Y Inverse is A111963.

%Y Factors as A007318*A111959.

%Y Column k=0 gives A002426.

%Y Cf. A026325.

%K nonn,tabl,easy

%O 0,4

%A _Paul Barry_, Aug 23 2005