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

0,3

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.

Links

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

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A033721 A033739 A033733 * A264954 A212085 A265882

Adjacent sequences: A115948 A115949 A115950 * A115952 A115953 A115954

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 14 2006

Status

approved