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

0,4

Comments

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

Links

Table of n, a(n) for n=0..65.

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)

Example

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)

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

Crossrefs

Row sums are A111961.

Diagonal sums are A111962.

Inverse is A111963.

Factors as A007318*A111959.

Column k=0 gives A002426.

Cf. A026325.

Sequence in context: A094531 A274293 A161009 * A130462 A373506 A059380

Adjacent sequences: A111957 A111958 A111959 * A111961 A111962 A111963

Keyword

nonn,tabl,easy

Author

Paul Barry, Aug 23 2005

Status

approved