A111960 Renewal array for central trinomial numbers A002426.
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, 1, 393, 547, 441, 251, 105, 33, 7, 1, 1107, 1640, 1428, 888, 420, 152, 42, 8, 1, 3139, 4921, 4572, 3076, 1596, 654, 210, 52, 9, 1, 8953, 14762, 14535, 10456, 5880, 2652, 966, 280, 63, 10, 1
Offset: 0
Examples
Triangle T(n,k) begins: 1; 1, 1; 3, 2, 1; 7, 7, 3, 1; 19, 20, 12, 4, 1; 51, 61, 40, 18, 5, 1; ... From _Paul Barry_, May 12 2009: (Start) Production matrix is 1, 1, 2, 1, 1, 0, 2, 1, 1, -2, 0, 2, 1, 1, 0, -2, 0, 2, 1, 1, 4, 0, -2, 0, 2, 1, 1. (End)
Crossrefs
Programs
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Maple
# Uses function PMatrix from A357368. Adds a row and column above and to the left. PMatrix(10, n -> A002426(n - 1)); # Peter Luschny, Oct 06 2022
Formula
Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
From Paul Barry, May 12 2009: (Start)
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)).
Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
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.
G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)
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