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 A037980 a(n) = (1/16)*( binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2 ). 2
 0, 1, 4, 109, 800, 19501, 168952, 3979830, 37566720, 862687045, 8615396504, 193710517650, 2015475061184, 44516469004294, 478043160040240, 10399216983867484, 114539008771344384, 2459029841101222485, 27657033766735102744, 586949749681986718650, 6719200545824895620800, 141147097812860184921810 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (see Identity (3.75) divided by four in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.) LINKS G. C. Greubel, Table of n, a(n) for n = 0..825 FORMULA From G. C. Greubel, Jun 22 2022: (Start) a(n)= A037976(n)/4. a(n) = (1/4)*Sum_{k=0..floor((n-1)/2)} binomial(2*n, 2*k+1)^2. a(n) = (1/16)*( (2*n+1)*A000108(2*n) - (-1)^n*(n+1)*A000108(n) + (1-(-1)^n)*(n+1)^2*A000108(n)^2 ). G.f.: (1/16)*(sqrt(1 + sqrt(1-16*x))/(sqrt(2)*sqrt(1-16*x)) - 1/sqrt(1+4*x)) + (1/(8*Pi))*( EllipticK(16*x) - EllipticK(-16*x)). (End) MAPLE A037980 := proc(n) binomial(4*n, 2*n) -(-1)^n*binomial(2*n, n)+(1-(-1)^n)*binomial(2*n, n)^2 ; %/16 ; end proc: # R. J. Mathar, Oct 20 2015 MATHEMATICA With[{B=Binomial}, Table[(1/16)*(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) [(1/16)*((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; [(1/16)*(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, A037976. Sequence in context: A259373 A076265 A114876 * A240626 A297916 A298541 Adjacent sequences: A037977 A037978 A037979 * A037981 A037982 A037983 KEYWORD nonn,easy,changed AUTHOR N. J. A. Sloane EXTENSIONS More terms added by G. C. Greubel, Jun 22 2022 STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)