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%I #25 Sep 08 2022 08:45:24
%S 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,
%T 924,0,0,0,0,280,2520,5544,3432,0,0,0,0,70,2520,13860,24024,12870,0,0,
%U 0,0,0,1260,18480,72072,102960,48620,0,0,0,0,0,252,13860,120120,360360,437580,184756
%N Expansion of 1/sqrt(1-4*x*y-4*x^2*y).
%C Row sums are A006139. Diagonal sums are A115962.
%C Coefficients of 2^n * P(n, x) with P the Legendre P polynomials. Reflection of triangle A008556. - _Ralf Stephan_, Apr 07 2016.
%H G. C. Greubel, <a href="/A115951/b115951.txt">Rows n = 0..100 of triangle, flattened</a>
%F Number triangle T(n,k) = C(2k,k)*C(k,n-k).
%F From _Peter Bala_, Sep 02 2015: (Start)
%F Binomial transform is A063007; equivalently, P * M = A063007, where P denotes Pascal's triangle A007318 and M denotes the present array.
%F P * M * P^-1 is a signed version of A063007. (End)
%e Triangle begins
%e 1,
%e 0, 2,
%e 0, 2, 6,
%e 0, 0, 12, 20,
%e 0, 0, 6, 60, 70,
%e 0, 0, 0, 60, 280, 252,
%e 0, 0, 0, 20, 420, 1260, 924
%t Table[Binomial[2k, k]Binomial[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Michael De Vlieger_, Sep 02 2015 *)
%o (Magma) /* As triangle */ [[Binomial(2*k,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 03 2015
%o (PARI) {T(n,k) = binomial(2*k,k)*binomial(k,n-k)}; \\ _G. C. Greubel_, May 06 2019
%o (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
%Y Cf. A006139 (row sums), A063007 (binomial transform), A115962 (diagonal sums).
%K easy,nonn,tabl
%O 0,3
%A _Paul Barry_, Mar 14 2006
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