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URL: https://oeis.org/A115951

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A115951
Expansion of 1/sqrt(1-4*x*y-4*x^2*y).
5
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.
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
(SageMath) [[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 A380977
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Mar 14 2006
STATUS
approved