A000172 The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3. (Formerly M1971 N0781)

1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596

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 floor((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 * binomial(2*k,n). Zhi-Wei Sun conjectured that for every n = 2,3,... the polynomial f_n(x) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*k,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

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

This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017

Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 2*x*y*z), 1/(1 - x - y - z + 4*x*y*z), 1/(1 + y + z + x*y + y*z + x*z + 2*x*y*z), 1/(1 + x + y + z + 2*(x*y + y*z + x*z) + 4*x*y*z). - Gheorghe Coserea, Jul 04 2018

a(n) is the constant term in the expansion of ((1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019

Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - x*y*z). - Seiichi Manyama, Jul 11 2020

Named after the Swiss mathematician Jérôme Franel (1859-1939). - Amiram Eldar, Jun 15 2021

It appears that a(n) is equal to the coefficient of (x*y*z)^n in the expansion of (1 + x + y - z)^n * (1 + x - y + z)^n * (1 - x + y + z)^n. Cf. A036917. - Peter Bala, Sep 20 2021

References

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.

Jérôme Franel, On a question of Laisant, Intermédiaire des Mathématiciens, vol 1 1894 pp 45-47

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 56.

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

John 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).

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)

Boris Adamczewski, Jason P. Bell and Eric Delaygue, Algebraic independence of G-functions and congruences à la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.

Prarit Agarwal and June Nahmgoong, Singlets in the tensor product of an arbitrary number of Adjoint representations of SU(3), arXiv:2001.10826 [math.RT], 2020.

Richard Askey, Orthogonal Polynomials and Special Functions, SIAM, 1975; see p. 43.

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

P. Barrucand, Problem 75-4, A Combinatorial Identity, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, Vol.. 294 (May 24 1982), pp. 657-660.

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} , arXiv:0712.3946 [math.CO], 2007.

David Callan, A combinatorial interpretation for an identity of Barrucand, JIS, Vol. 11 (2008), Article 08.3.4.

Marc Chamberland and Armin Straub, Apéry Limits: Experiments and Proofs, arXiv:2011.03400 [math.NT], 2020.

Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.

M. Coster, Email, Nov 1990

T. W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory, Series A, Vol. 52, No. 1 (1989), pp. 77-83.

Eric Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.

Robert W. Donley Jr, Directed path enumeration for semi-magic squares of size three, arXiv:2107.09463 [math.CO], 2021.

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

Carsten Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., VOl. 43, No. 1 (2005), pp. 31-45.

Jeff D. Farmer and Steven C. Leth, An asymptotic formula for powers of binomial coefficients, Math. Gaz., Vol. 89, No. 516 (2005), pp. 385-391.

Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See A p. 2.

Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).

S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.

Nick Hobson, Python program for this sequence.

Bradley Klee, Checking Weierstrass data, 2023.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 282.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, Vol. 2, No. 5 (2016).

Guo-Shuai Mao, Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers, arXiv:2111.08778 [math.NT], 2021.

Guo-Shuai Mao and Yan Liu, On a congruence conjecture of Z.-W. Sun involving Franel numbers, arXiv:2111.08775 [math.NT], 2021.

Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.

Marci A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory, Vo. 27 (1987), pp. 304-309.

Juan Pla, Problem H-505, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 33, No. 5 (1995), p. 473; Sum Formulae!, Solution to Problem H-505 by Paul S. Bruckman, ibid., Vol. 35, No. 1 (1997), pp. 93-95.

Armin Straub, and Wadim Zudilin, Sums of powers of binomials, their Apéry limits, and Franel's suspicions, arXiv:2112.09576 [math.NT], 2021.

Volker Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.

Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.

Zhi-Wei Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011.

Zhi-Wei Sun, Connections between p = x^2+3y^2 and Franel numbers, J. Number Theory, Vol. 133 (2013), pp. 2919-2928.

Zhi-Wei Sun, Conjectures involving arithmetical sequences, arXiv:1208.2683v9 [math.CO] 2013; 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.

Zhi-Wei Sun, Congruences involving g_n(x) = Sum_{k= 0..n} C(n,k)^2 C(2k,k) x^k, arXiv preprint arXiv:1407.0967 [math.NT], 2014.

Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arxiv 1401.5400 [math.CO], 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.

Don Zagier, Integral solutions of Apéry-like recurrence equations. See line A in sporadic solutions table of page 5.

Bao-Xuan Zhu, Higher order log-monotonicity of combinatorial sequences, arXiv preprint arXiv:1309.6025 [math.CO], 2013.

Formula

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

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

D-finite with recurrence (n + 1)^2*a(n+1) = (7*n^2 + 7*n + 2)*a(n) + 8*n^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} (3*n)!/n!^3 * x^(2*n)/(1 - 2*x)^(3*n+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)),

with 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(2*k,n)*C(2*k,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

For r a nonnegative integer, Sum_{k = r..n} C(k,r)^3*C(n,k)^3 = C(n,r)^3*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016

a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^2. - Peter Luschny, May 31 2017

From Gheorghe Coserea, Jul 04 2018: (Start)

a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 2^(n-2*k).

G.f. y=A(x) satisfies: 0 = x*(x + 1)*(8*x - 1)*y'' + (24*x^2 + 14*x - 1)*y' + 2*(4*x + 1)*y. (End)

a(n) = [x^n] (1 - x^2)^n*P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 56. - Peter Bala, Mar 24 2022

a(n) = (2^n/(4*Pi^2)) * Integral_{x,y=0..2*Pi} (1+cos(x)+cos(y)+cos(x+y))^n dx dy = (8^n/(Pi^2)) * Integral_{x,y=0..Pi} (cos(x)*cos(y)*cos(x+y))^n dx dy (Pla, 1995). - Amiram Eldar, Jul 16 2022

a(n) = Sum_{k = 0..n} m^(n-k)*binomial(n,k)*binomial(n+2*k,n)*binomial(2*k,k) at m = -4. Cf. A081798 (m = 1), A006480 (m = 0), A124435 (m = -1), A318109 (m = -2) and A318108 (m = -3). - Peter Bala, Mar 16 2023

From Bradley Klee, Jun 05 2023: (Start)

The g.f. T(x) obeys a period-annihilating ODE:

0=2*(1 + 4*x)*T(x) + (-1 + 14*x + 24*x^2)*T'(x) + x*(1 + x)*(-1 + 8*x)*T''(x).

The periods ODE can be derived from the following Weierstrass data:

g2 = (4/243)*(1 - 8*x + 240*x^2 - 464*x^3 + 16*x^4);

g3 = -(8/19683)*(1 - 12*x - 480*x^2 + 3080*x^3 - 12072*x^4 + 4128*x^5 +

64*x^6);

which determine an elliptic surface with four singular fibers. (End)

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
A000172_list := proc(len) series(hypergeom([], [1, 1], x)^2, x, len);
seq((n!)^3*coeff(%, x, n), n=0..len-1) end:
A000172_list(21); # Peter Luschny, May 31 2017

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()
[next(a) for i in range(21)]   # Peter Luschny, Oct 12 2013
(PARI) A000172(n)={sum(k=0, (n-1)\2, binomial(n, k)^3)*2+if(!bittest(n, 0), binomial(n, n\2)^3)} \\ M. F. Hasler, Sep 21 2015

Crossrefs

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

Cf. A181543, A006480, A141057, A000578, A007318.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

Sequence in context: A323935 A165817 A243644 * A097971 A191277 A290443

Adjacent sequences: A000169 A000170 A000171 * A000173 A000174 A000175

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved