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 A037976 a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2). 2
 0, 4, 16, 436, 3200, 78004, 675808, 15919320, 150266880, 3450748180, 34461586016, 774842070600, 8061900244736, 178065876017176, 1912172640160960, 41596867935469936, 458156035085377536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (See (3.75) in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.) LINKS G. C. Greubel, Table of n, a(n) for n = 0..825 FORMULA a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2). From G. C. Greubel, Jun 22 2022: (Start) a(n) = Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2. a(n) = (1/4)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n*(n+1)^2*A000108(n)^2 ). G.f.: (1/4)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(2*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End) MAPLE A037976 := proc(n) binomial(4*n, 2*n)/4-(-1)^n*binomial(2*n, n)/4+(1-(-1)^n)*binomial(2*n, n)^2/4 ; end proc: seq(A037976(n), n=0..30) ; # R. J. Mathar, Jul 26 2015 MATHEMATICA With[{B=Binomial}, Table[(1/4)*(B[4*n, 2*n] +B[2*n, n]^2 -2*(-1)^n*B[B[2*n, n] +1, 2]), {n, 0, 30}]] (* G. C. Greubel, Jun 22 2022 *) PROG (Magma) [(1/4)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n) +(1-(-1)^n)*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022 (SageMath) b=binomial; [(1/4)*(b(4*n, 2*n) -(-1)^n*b(2*n, n) +(1-(-1)^n)*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022 CROSSREFS Cf. A000108, A037964, A037972. Sequence in context: A156337 A000874 A061580 * A013138 A262546 A013006 Adjacent sequences: A037973 A037974 A037975 * A037977 A037978 A037979 KEYWORD nonn AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 30 21:59 EDT 2023. Contains 365812 sequences. (Running on oeis4.)