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

0,5

Comments

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

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

Links

Table of n, a(n) for n=0..54.

Formula

T(n,n) = 1 = A000012(n); T(n+1,n) = 2n = A005843(n); T(n+2,n) = 2n*(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(1-4x)/(sqrt(1-4x)-y*x).

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

Example

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;

Maple

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

Crossrefs

Cf. A054335 and columns listed there.

Sequence in context: A011312 A275328 A147720 * A127631 A122538 A090238

Adjacent sequences: A205810 A205811 A205812 * A205814 A205815 A205816

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 01 2012

Status

approved