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A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n.
(Formerly M1971 N0781)
62
1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Cusick gives a general method of deriving recurrences for the r-th order Franel numbers (this is the sequence of third-order Franel numbers), with [(r+3)/2] terms.

This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004

a(1) = 2 is the only prime Franel number. Semiprime Franel numbers include: a(2) = 10 = 2 * 5, a(4) = 346 = 2 * 173, a(8) = 739162 = 2 * 369581. - Jonathan Vos Post, May 22 2005

An identity of V. Strehl states that a(n) = sum_{k=0}^n C(n,k)^2 * C(2k,n). Zhi-Wei Sun conjectured that for every n=2,3,... the polynomial f_n(x) = sum_{k=0}^n C(n,k)^2 * C(2k,n) * x^{n-k} is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013

Conjecture: a(n)== 2 (mod n^3) iff n is prime. - Gary Detlefs, Mar 22 2013

It is trivial that a(p) == 2 (mod p^3) for any prime p since p | C(p,k) for all k = 1,...,p-1. - Zhi-Wei Sun, Aug 14 2013

a(n) is the maximal number of totally mixed Nash equilibria in games of 3 players, each with n+1 pure options. - Raimundas Vidunas, Jan 22 2014

REFERENCES

R. Askey, Orthognal Polynomials and Special Functions, SIAM, 1975; see p. 43.

P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., 17 (1975), 168. Solution by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev. 18 (1976), 303.

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, May 24 1982.

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

T. W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory, A 52 (1989), 77-83.

Doslic, Tomislav and Veljan, Darko. Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From N. J. A. Sloane, May 01 2012

C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.

J. Franel, Intermediaire des Mathematiciens, 1894.

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.

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).

Z.-W. Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. the 6th China-Japan Sem. (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258; http://math.nju.edu.cn/~zwsun/142p.pdf.

Z.-W. Sun, Connections between p = x^2+ 3y^2 and Franel numbers, J. Number Theory 133 (2013), no. 9, 2914-2928.

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

David Callan, A combinatorial interpretation for the identity Sum_{k=0}^{n} binom{n}{k} Sum_{j=0}^{k} binom{k}{j}^{3}= Sum_{k=0}^{n} binom{n}{k}^{2}binom{2k}{k} .

E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131, 2013

Jeff D. Farmer, Steven C. Leth, An asymptotic formula for powers of binomial coefficients, Math. Gaz. 89 (516) (2005) 385-391.

Nick Hobson, Python program for this sequence

V. Strehl, Recurrences and Legendre transform

Z.-W. Sun, Congruences for Franel numbers, Arxiv preprint arXiv:1112.1034, 2011.

R. Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arxiv 1401.5400 (2014)

Eric Weisstein's World of Mathematics, Binomial Sums

Eric Weisstein's World of Mathematics, Franel Number

Eric Weisstein's World of Mathematics, Schmidt's Problem

Bao-Xuan Zhu, Higher order log-monotonicity of combinatorial sequences, arXiv preprint arXiv:1309.6025, 2013

FORMULA

A002893(n) = Sum_{m=0..n} binomial(n, m) a(m) [Barrucand]

Sum C(n, k)^3, k=0..n = (-1)^n Integral_{0..infinity} L_k(x)^3 exp(-x) dx. - from Askey's book, p. 43

(n+1)^2 * a(n+1) = (7n^2+7n+2) * a(n) + 8n^2 * a(n-1) [Franel]. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 31 2001

a(n) ~ 2*3^(-1/2)*pi^-1*n^-1*2^(3*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002

O.g.f.: A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(2n)/(1-2x)^(3n+1). - Paul D. Hanna, Oct 30 2010

G.f.: hypergeom([1/3, 2/3], [1], 27 x^2 / (1 - 2x)^3) / (1 - 2x). - Michael Somos, Dec 17 2010

G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = [ Sum_{n>=0} x^n/n!^3 ]^2. - Paul D. Hanna, Jan 19 2011

G.f.: A(x)= 1/(1-2*x)*(1+6*(x^2)/(G(0)-6*x^2));

G(k)=3*(x^2)*(3*k+1)*(3*k+2) + ((1-2*x)^3)*((k+1)^2) - 3*(x^2)*((1-2*x)^3)*((k+1)^2)*(3*k+4)*(3*k+5)/G(k+1) ;  (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011

In 2011 Zhi-Wei Sun found the formula sum_{k=0}^n C(2k,n)*C(2k,k)*C(2(n-k),n-k) = 2^n*a(n) and proved it via the Zeilberger algorithm. - Zhi-Wei Sun, Mar 20 2013

0 = a(n)*(a(n+1)*(-2048*a(n+2) - 3392*a(n+3) + 768*a(n+4)) + a(n+2)*(-1280*a(n+2) - 2912*a(n+3) + 744*a(n+4)) + a(n+3)*(+288*a(n+3) - 96*a(n+4))) + a(n+1)*(a(n+1)*(-704*a(n+2) - 1232*a(n+3) + 288*a(n+4)) + a(n+2)*(-560*a(n+2) - 1372*a(n+3) + 364*a(n+4)) + a(n+3)*(+154*a(n+3) - 53*a(n+4))) + a(n+2)*(a(n+2)*(+24*a(n+2) + 70*a(n+3) - 20*a(n+4)) + a(n+3)*(-11*a(n+3) + 4*a(n+4))) for all n in Z. - Michael Somos, Jul 16 2014

EXAMPLE

O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 56*x^3 + 346*x^4 + 2252*x^5 +...

O.g.f.: A(x) = 1/(1-2*x) + 3!*x^2/(1-2*x)^4 + (6!/2!^3)*x^4/(1-2*x)^7 + (9!/3!^3)*x^6/(1-2*x)^10 + (12!/4!^3)*x^8/(1-2*x)^13 +... - Paul D. Hanna, Oct 30 2010

Let g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^3, then

A(x) = 1 + 2*x + 10*x^2/2!^3 + 56*x^3/3!^3 + 346*x^4/4!^3 +... where

A(x) = [1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +...]^2. - Paul D. Hanna

MAPLE

A000172 := proc(n)

    add(binomial(n, k)^3, k=0..n) ;

end proc:

seq(A000172(n), n=0..10) ; # R. J. Mathar, Jul 26 2014

MATHEMATICA

Table[Sum[Binomial[n, k]^3, {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Aug 24 2011 *)

Table[ HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1], {n, 0, 20}]  (* Jean-François Alcover, Jul 16 2012, after symbolic sum *)

a[n_] := Sum[ Binomial[2k, n]*Binomial[2k, k]*Binomial[2(n-k), n-k], {k, 0, n}]/2^n; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 20 2013, after Zhi-Wei Sun *)

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/3, 2/3, 1, 27 x^2 / (1 - 2 x)^3] / (1 - 2 x), {x, 0, n}]; (* Michael Somos, Jul 16 2014 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)/(1-2*x+x*O(x^n))^(3*m+1)), n)} \\ Paul D. Hanna, Oct 30 2010

(PARI) {a(n)=n!^3*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)} \\ Paul D. Hanna, Jan 19 2011

(Haskell)

a000172 = sum . map a000578 . a007318_row

-- Reinhard Zumkeller, Jan 06 2013

(Sage)

def A000172():

    x, y, n = 1, 2, 1

    while true:

        yield x

        n += 1

        x, y = y, (8*(n-1)^2*x + (7*n^2-7*n + 2)*y) / n^2

a = A000172()

[a.next() for i in range(21)]   # Peter Luschny, Oct 12 2013

CROSSREFS

Cf. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.

Cf. A181543, A006480.

Cf. A141057.

Cf. A000578, A007318.

Sequence in context: A108490 A165817 A243644 * A097971 A191277 A093303

Adjacent sequences:  A000169 A000170 A000171 * A000173 A000174 A000175

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 1 07:07 EDT 2014. Contains 246288 sequences.