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A002457 a(n) = (2n+1)!/n!^2.
(Formerly M4198 N1752)
105
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

Sum_{n>=0} 1/a(n) = 2*Pi/3^(3/2). - Jaume Oliver Lafont, Mar 07 2009

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-by-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

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).

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21.

Wallis, J., Operum Mathematicorum, pars altera, Oxford, 1656, pp 31,34 [Marc van Leeuwen, Apr 14 2010]

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

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]

R. Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117.

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.

P.-Y. Huang, S.-C. Liu, Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

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.

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(binomial(n-1+i, i)*(n-i)/2^(n-1+i), i=0..n-1).

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 * binomial(n, i)^2, i=0.. n) = 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<=k<=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

a(n) = sum{k=0..n, C(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],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

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; (* 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 */

(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.

Sequence in context: A125316 A092439 A082149 * A137400 A220830 A199938

Adjacent sequences:  A002454 A002455 A002456 * A002458 A002459 A002460

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified December 7 17:16 EST 2016. Contains 278890 sequences.