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

Author

Paul Barry, Aug 23 2005

Keywords

• nonn
• tabl
• easy

Comments

Also the convolution triangle of A002426. - Peter Luschny, Oct 06 2022

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

Row sums are A111961.

Diagonal sums are A111962.

Inverse is A111963.

Factors as A007318*A111959.

Column k=0 gives A002426.

Cf. A026325.

Programs

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)