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 A002457 a(n) = (2n+1)!/n!^2. (Formerly M4198 N1752) 130
 1, 6, 30, 140, 630, 2772, 12012, 51480, 218790, 923780, 3879876, 16224936, 67603900, 280816200, 1163381400, 4808643120, 19835652870, 81676217700, 335780006100, 1378465288200, 5651707681620, 23145088600920, 94684453367400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Expected number of matches remaining in Banach's modified matchbox problem (counted when last match is drawn from one of the two boxes), multiplied by 4^(n-1). - Michael Steyer, Apr 13 2001 Hankel transform is (-1)^n*A014480(n). - Paul Barry, Apr 26 2009 Convolved with A000108: (1, 1, 1, 5, 14, 42, ...) = A000531: (1, 7, 38, 187, 874, ...). - Gary W. Adamson, May 14 2009 Convolution of A000302 and A000984. - Philippe Deléham, May 18 2009 1/a(n) is the integral of (x(1-x))^n on interval [0,1]. Apparently John Wallis computed these integrals for n=0,1,2,3,.... A004731, shifted left by one, gives numerators/denominators of related integrals (1-x^2)^n on interval [0,1]. - Marc van Leeuwen, Apr 14 2010 Extend the triangular peaks of Dyck paths of semilength n down to the baseline forming (possibly) larger and overlapping triangles. a(n) = sum of areas of these triangles. Also a(n) = triangular(n) * Catalan(n). - David Scambler, Nov 25 2010 Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n of B equals a(n-1). - T. D. Noe, May 01 2011 Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+1. - David Scambler, Apr 29 2013 Denominator of central elements of Leibniz's Harmonic Triangle A003506. Central terms of triangle A116666. - Reinhard Zumkeller, Nov 02 2013 Number of distinct strings of length 2n+1 using n letters A, n letters B, and 1 letter C. - Hans Havermann, May 06 2014 Number of edges in the Hasse diagram of the poset of partitions in the n X n box ordered by containment (from Havermann's comment above, C represents the square added in the edge). - William J. Keith, Aug 18 2015 Let V(n, r) denote the volume of an n-dimensional sphere with radius r then V(n, 1/2^n) = V(n-1, 1/2^n) / a((n-1)/2) for all odd n. - Peter Luschny, Oct 12 2015 Sum_{n >= 0} 2^(n+1)/a(n) = Pi, related to Newton/Euler's Pi convergence transformation series. - Tony Foster III, Jul 28 201. See the Weisstein Pi link, eq. (23). - Wolfdieter Lang, Aug 26 2016 a(n) = the result of processing the n+1 row of Pascal's triangle A007318 with the method of A067056. Example: Let n=3. Given the 4th row of Pascal's triangle 1,4,6,4,1, we get 1*(4+6+4+1) + (1+4)*(6+4+1) + (1+4+6)*(4+1) + (1+4+6+4)*1 = 15+55+55+15 = 140 = a(3). - J. M. Bergot, May 26 2017 4^n/a(n) is the integral of (1 - x^2)^n on interval [0, 1]. - Michael Somos, Jun 13 2019 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 159. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25; p. 168, #30. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I. C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 449. M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129. C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514. A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992. J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). J. Wallis, Operum Mathematicorum, pars altera, Oxford, 1656, pp 31,34 [Marc van Leeuwen, Apr 14 2010] LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000[Terms 0 to 200 computed by T. D. Noe; terms 201 to 1000 by G. C. Greubel, Jan 14 2017] Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020. W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy] Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.See p. 15, Remark 4.2 R. Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117. Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43. F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, A closed formula for the number of convex permutominoes, arXiv:math/0702550 [math.CO], 2007. Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5. Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020). P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45. Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only] C. Lanczos, Applied Analysis (Annotated scans of selected pages) A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013. H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135. H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy] J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466. Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 5. T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21. Eric Weisstein's World of Mathematics, Central Beta Function Eric Weisstein's World of Mathematics, Pi Formulas Y. Q. Zhao, Introduction to Probability with Applications FORMULA G.f.: (1-4x)^(-3/2) = 1F0(3/2;;4x). a(n-1) = binomial(2*n, n)*n/2 = binomial(2*n-1, n)*n. a(n-1) = 4^(n-1)*Sum_{i=0..n-1} binomial(n-1+i, i)*(n-i)/2^(n-1+i). a(n) ~ 2*Pi^(-1/2)*n^(1/2)*2^(2*n)*{1 + 3/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 21 2001 (2*n+2)!/(2*n!*(n+1)!) = (n+n+1)!/(n!*n!) = 1/beta(n+1, n+1) in A061928. Sum_{i=0..n} i * binomial(n, i)^2 = n*binomial(2*n, n)/2. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000 a(n) ~ 2*Pi^(-1/2)*n^(1/2)*2^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002 a(n) = 1/Integral_{x=0..1} x^n (1-x)^n dx. - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 10 2003 E.g.f.: exp(2*x)*((1+4*x)*BesselI(0, 2*x)+4*x*BesselI(1, 2*x)). - Vladeta Jovovic, Sep 22 2003 a(n) = Sum_{i+j+k=n} binomial(2i, i)*binomial(2j, j)*binomial(2k, k). - Benoit Cloitre, Nov 09 2003 a(n) = (2*n+1)*A000984(n) = A005408(n)*A000984(n). - Zerinvary Lajos, Dec 12 2010 a(n-1) = Sum_{k=0..n} A039599(n,k)*A000217(k), for n >= 1. - Philippe Deléham, Jun 10 2007 Sum of (n+1)-th row terms of triangle A132818. - Gary W. Adamson, Sep 02 2007 Sum_{n>=0} 1/a(n) = 2*Pi/3^(3/2). - Jaume Oliver Lafont, Mar 07 2009 a(n) = Sum_{k=0..n} binomial(2k,k)*4^(n-k). - Paul Barry, Apr 26 2009 a(n) = A000217(n) * A000108(n). - David Scambler, Nov 25 2010 a(n) = f(n, n-3) where f is given in A034261. a(n) = A005430(n+1)/2 = A002011(n)/4. a(n) = binomial(2n+2, 2) * binomial(2n, n) / binomial(n+1, 1), a(n) = binomial(n+1, 1) * binomial(2n+2, n+1) / binomial(2, 1) = binomial(2n+2, n+1) * (n+1)/2. - Rui Duarte, Oct 08 2011 G.f.: (G(0) - 1)/(4*x) where G(k) = 1 + 2*x*((2*k + 3)*G(k+1) - 1)/(k + 1). - Sergei N. Gladkovskii, Dec 03 2011 [Edited by Michael Somos, Dec 06 2013] G.f.: 1 - 6*x/(G(0)+6*x) where G(k)= 1 + (4*x+1)*k - 6*x - (k+1)*(4*k-2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 13 2012 G.f.: Q(0), where Q(k)= 1 + 4*(2*k + 1)*x*(2*k + 2 + Q(k+1))/(k+1). - Sergei N. Gladkovskii, May 10 2013 [Edited by Michael Somos, Dec 06 2013] G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 4*x*(2*k+3)/(4*x*(2*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013 a(n) = 2^(4n)/Sum_{k=0..n} (-1)^k*C(2n+1,n-k)/(2k+1). - Mircea Merca, Nov 12 2013 a(n) = (2*n)!*[x^(2*n)] HeunC(0,0,-2,-1/4,7/4,4*x^2) where [x^n] f(x) is the coefficient of x^n in f(x) and HeunC is the Heun confluent function. - Peter Luschny, Nov 22 2013 0 = a(n) * (16*a(n+1) - 2*a(n+2)) + a(n+1) * (a(n+2) - 6*a(n+1)) for all n in Z. - Michael Somos, Dec 06 2013 a(n) = 4^n*binomial(n+1/2, 1/2). - Peter Luschny, Apr 24 2014 a(n) = 4^n*hypergeom([-2*n,-2*n-1,1/2],[-2*n-2,1],2)*(n+1)*(2*n+1). - Peter Luschny, Sep 22 2014 a(n) = 4^n*hypergeom([-n,-1/2],,1). - Peter Luschny, May 19 2015 a(n) = 2*4^n*Gamma(3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015 Boas-Buck recurrence: a(n) = (6/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, and a(0) = 1. Proof from a(n) = A046521(n+1, 1). See a comment in A046521. - Wolfdieter Lang, Aug 10 2017 a(n) = (1/3)*Sum_{i = 0..n+1} C(n+1,i)*C(n+1,2*n+1-i)*C(3*n+2-i,n+1) = (1/3)*Sum_{i = 0..2*n+1} (-1)^(i+1)*C(2*n+1,i)*C(n+i+1,i)^2. - Peter Bala, Feb 07 2018 a(n) = (2*n+1)*binomial(2*n, n). - Kolosov Petro, Apr 16 2018 a(n) = (-4)^n*binomial(-3/2, n). - Peter Luschny, Oct 23 2018 a(n) = 1 / Sum_{s=0..n} {(-1)^s * binomial(n, s) / (n+s+1)}. - Kolosov Petro, Jan 22 2019 a(n) = Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k). - Peter Bala, Feb 25 2019 D-finite with recurrence: 0 = a(n)*(6 + 4*n) - a(n+1)*(n + 1) for all n in Z. - Michael Somos, Jun 13 2019 Sum_{n>=0} (-1)^n/a(n) = 4*arcsinh(1/2)/sqrt(5). - Amiram Eldar, Sep 10 2020 EXAMPLE G.f. = 1 + 6*x + 30*x^2 + 140*x^3 + 630*x^4 + 2772*x^5 + 12012*x^6 + 51480*x^7 + ... MAPLE A002457:=n->(n+1) * binomial(2*(n+1), (n+1)) / 2; seq(A002457(n), n=0..50); seq((2*n)!*coeff(series(HeunC(0, 0, -2, -1/4, 7/4, 4*x^2), x, 2*n+1), x, 2*n), n=0..22); # Peter Luschny, Nov 22 2013 MATHEMATICA a[n_]:=(2*n+1)!/n!^2; Array[f, 23, 0] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *) PROG (PARI) {a(n) = if( n<0, 0, (2*n + 1)! / n!^2)} /* Michael Somos, Dec 09 2002 */ (PARI) a(n) = (2*n+1)*binomial(2*n, n); \\ Altug Alkan, Apr 16 2018 (Haskell) a002457 n = a116666 (2 * n + 1) (n + 1) -- Reinhard Zumkeller, Nov 02 2013 (Sage) A002457 = lambda n: binomial(n+1/2, 1/2)<<2*n [A002457(n) for n in range(23)] # Peter Luschny, Sep 22 2014 (MAGMA) [Factorial(2*n+1)/Factorial(n)^2: n in [0..25]]; // Vincenzo Librandi, Oct 12 2015 CROSSREFS Cf. A000531 (Banach's original match problem). Cf. A033876, A000984, A001803, A132818, A046521 (second column). A diagonal of A331430. The rightmost diagonal of the triangle A331431. Sequence in context: A125316 A092439 A082149 * A137400 A220830 A199938 Adjacent sequences:  A002454 A002455 A002456 * A002458 A002459 A002460 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 19 21:59 EDT 2021. Contains 348095 sequences. (Running on oeis4.)