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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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30
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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 (jvospost3(AT)gmail.com), May 22 2005
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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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
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
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} .
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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). [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 30 2010]
G.f.: hypergeom([1/3, 2/3], [1], 27 x^2 / (1 - 2x)^3) / (1 - 2x) [From Michael Somos, Dec 17 2010]
G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = [ Sum_{n>=0} x^n/n!^3 ]^2. [From Paul D. Hanna (pauldhanna(AT)juno.com), 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
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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 +... [From Paul D. Hanna (pauldhanna(AT)juno.com), 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
| Table[Sum[Binomial[n, k]^3, {k, 0, n}], {n, 0, 30}] (* From Harvey P. Dale, Aug 24 2011 *)
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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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 30 2010]
(PARI) {a(n)=n!^3*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)} [From Paul D. Hanna, Jan 19 2011]
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CROSSREFS
| Cf. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.
Cf. A181543, A006480. [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 30 2010]
Cf. A141057.
Sequence in context: A122826 A108490 A165817 * A097971 A191277 A093303
Adjacent sequences: A000169 A000170 A000171 * A000173 A000174 A000175
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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