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%I #53 Dec 15 2023 17:41:07
%S 1,1,2,1,4,6,1,6,15,20,1,8,28,56,70,1,10,45,120,210,252,1,12,66,220,
%T 495,792,924,1,14,91,364,1001,2002,3003,3432,1,16,120,560,1820,4368,
%U 8008,11440,12870,1,18,153,816,3060,8568,18564,31824,43758,48620
%N Triangle of binomial(2*n, k) with n >= k.
%C From _Wolfdieter Lang_, Sep 19 2012: (Start)
%C The triangle a(n,k) appears in the formula F(2*l+1)^(2*n) = (sum(a(n,k)*L(2*(n-k)*(2*l+1)),k=0..n-1) + a(n,n))/5^n, n>=0, l>=0, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
%C The signed triangle as(n,k):=a(n,k)*(-1)^k appears in the formula F(2*l)^(2*n) = (sum(as(n,k)*L(4*(n-k)*l),k=0..n-1) + as(n,n))/5^n, n>=0, l>=0. Proof with the Binet-de Moivre formula for F and L and the binomial formula. (End)
%H G. C. Greubel, <a href="/A062344/b062344.txt">Rows n = 0..100 of triangle, flattened</a>
%H E. H. M. Brietzke, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.050">An identity of Andrews and a new method for the Riordan array proof of combinatorial identities</a>, Discrete Math., 308 (2008), 4246-4262.
%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F a(n,k) = a(n,k-1)*((2n+1)/k-1) with a(n,0)=1.
%F G.f.: 1/((1-sqrt(1-4*x*y))^4/(16*x*y^2) + sqrt(1-4*x*y) - x). - _Vladimir Kruchinin_, Jan 26 2021
%e Rows start
%e (1),
%e (1,2),
%e (1,4,6),
%e (1,6,15,20)
%e etc.
%e Row n=2, (1,4,6):
%e F(2*l+1)^4 = (1*L(4*(2*l+1)) + 4*L(2*(2*l+1)) + 6)/25,
%e F(2*l)^4 = (1*L(8*l) - 4*L(4*l) + 6)/25, l>=0, F=A000045, L=A000032. See a comment above. - _Wolfdieter Lang_, Sep 19 2012
%t Flatten[Table[Binomial[2 n, k], {n, 0, 20}, {k, 0, n}]] (* _G. C. Greubel_, Jun 28 2018 *)
%o (Maxima) create_list(binomial(2*n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */
%o (PARI) for(n=0, 20, for(k=0, n, print1(binomial(2*n, k), ", "))) \\ _G. C. Greubel_, Jun 28 2018
%o (Magma) [[Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // _G. C. Greubel_, Jun 28 2018
%Y Columns include (sometimes truncated) A000012, A005843, A000384, A002492, A053134 etc. Right hand side includes A000984, A001791, A002694, A002696 etc. Row sums are A032443. Row alternate differences (e.g., 6-4+1=3 or 20-15+6-1=10) are A001700.
%Y Cf. A122366.
%Y a(2n,n) gives A005810.
%K nonn,tabl
%O 0,3
%A _Henry Bottomley_, Jul 06 2001
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