A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 20, 16, 6, 1, 0, 70, 64, 30, 8, 1, 0, 252, 256, 140, 48, 10, 1, 0, 924, 1024, 630, 256, 70, 12, 1, 0, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 0, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1

Offset: 0

Author

Philippe Deléham, Feb 01 2012

Keywords

• easy
• nonn
• tabl

Comments

Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).

T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - Peter Luschny, Oct 19 2022

Examples

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   6,   4,   1;
  0,  20,  16,   6,   1;
  0,  70,  64,  30,   8,   1;
  0, 252, 256, 140,  48,  10,   1;
		

Crossrefs

Cf. A054335 and columns listed there.

Programs

Maple

# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

Formula

T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).

Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.

G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).

Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.