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A000172
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Franel number a(n) = Sum C(n,k)^3, k=0..n.
(Formerly M1971 N0781)
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52
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1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596
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OFFSET
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0,2
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COMMENTS
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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.
Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion of a special point on a curve described by Beauville.
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
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REFERENCES
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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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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, Congruences for Franel numbers, Arxiv preprint arXiv:1112.1034, 2011
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.
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LINKS
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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} .
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
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
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FORMULA
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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]
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EXAMPLE
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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 +... [From 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. [From Paul D. Hanna]
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.
Cf. A181543, A006480. [Paul D. Hanna, Oct 30 2010]
Cf. A141057.
Cf. A000578, A007318.
Sequence in context: A122826 A108490 A165817 * A097971 A191277 A093303
Adjacent sequences: A000169 A000170 A000171 * A000173 A000174 A000175
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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