A115951 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).

1, 0, 2, 0, 2, 6, 0, 0, 12, 20, 0, 0, 6, 60, 70, 0, 0, 0, 60, 280, 252, 0, 0, 0, 20, 420, 1260, 924, 0, 0, 0, 0, 280, 2520, 5544, 3432, 0, 0, 0, 0, 70, 2520, 13860, 24024, 12870, 0, 0, 0, 0, 0, 1260, 18480, 72072, 102960, 48620, 0, 0, 0, 0, 0, 252, 13860, 120120, 360360, 437580, 184756

Offset: 0

Author

Paul Barry, Mar 14 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A006139. Diagonal sums are A115962.

Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - Ralf Stephan, Apr 07 2016.

Examples

			Triangle begins
   1,
   0,  2,
   0,  2,  6,
   0,  0, 12,  20,
   0,  0,  6,  60,  70,
   0,  0,  0,  60, 280,  252,
   0,  0,  0,  20, 420, 1260, 924
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).

Programs

Magma

/* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

Mathematica

Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Michael De Vlieger, Sep 02 2015 *)

PARI

{T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ G. C. Greubel, May 06 2019

Sage

[[binomial(2*k,k)*binomial(k,n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2019

Formula

Number triangle T(n,k) = C(2k,k)*C(k,n-k).

From Peter Bala, Sep 02 2015: (Start)

Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.

P * M * P^-1 is a signed version of A063007. (End)