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 A062344 Triangle of binomial(2*n, k) with n >= k. 7
 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, 495, 792, 924, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Wolfdieter Lang, Sep 19 2012: (Start) 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). 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) LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262. C. Lanczos, Applied Analysis (Annotated scans of selected pages) FORMULA a(n,k) = a(n,k-1)*((2n+1)/k-1) with a(n,0)=1. EXAMPLE Rows start   (1),   (1,2),   (1,4,6),   (1,6,15,20)   etc. Row n=2, (1,4,6): F(2*l+1)^4 = (1*L(4*(2*l+1)) + 4*L(2*(2*l+1)) + 6)/25, 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 MATHEMATICA Flatten[Table[Binomial[2 n, k], {n, 0, 20}, {k, 0, n}]] (* G. C. Greubel, Jun 28 2018 *) PROG (maxima) create_list(binomial(2*n, k), n, 0, 12, k, 0, n); [Emanuele Munarini, Mar 11 2011] (PARI) for(n=0, 20, for(k=0, n, print1(binomial(2*n, k), ", "))) \\ G. C. Greubel, Jun 28 2018 (MAGMA) [[Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Jun 28 2018 CROSSREFS 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. Cf. A122366. a(2n,n) gives A005810. Sequence in context: A033884 A208915 A199704 * A208759 A033877 A059369 Adjacent sequences:  A062341 A062342 A062343 * A062345 A062346 A062347 KEYWORD nonn,tabl AUTHOR Henry Bottomley, Jul 06 2001 STATUS approved

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Last modified July 9 14:56 EDT 2020. Contains 335543 sequences. (Running on oeis4.)