%I A000172 M1971 N0781
%S A000172 1,2,10,56,346,2252,15184,104960,739162,5280932,38165260,278415920,
%T A000172 2046924400,15148345760,112738423360,843126957056,6332299624282,
%U A000172 47737325577620,361077477684436,2739270870994736,20836827035351596
%N A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n.
%C A000172 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.
%C A000172 Comment from Matthijs Coster, Apr 28, 2004: This is the Taylor expansion 
               of a special point on a curve described by Beauville.
%C A000172 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
%C A000172 Number of permutations of 3 distinct letters (ABCD) each with n copies 
               such that free fixed points. E.g. if AAAAABBBBBCCCCC n=3*5 letters 
               permutations then free fixed points n5=2252 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 02 2006
%D A000172 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000172 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000172 R. Askey, Orthognal Polynomials and Special Functions, SIAM, 1975; see 
               p. 43.
%D A000172 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.
%D A000172 Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre 
               fibres singulieres, Comptes Rendus, Academie Science Paris, no. 294, 
               May 24 1982.
%D A000172 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], 
               Master's Thesis (unpublished), Aug 26 1983.
%D A000172 T. W. Cusick, Recurrences for sums of powers of binomial coefficients, 
               J. Combin. Theory, A 52 (1989), 77-83.
%D A000172 C. Elsner, On recurrence formulae for sums involving binomial coefficients, 
               Fib. Q., 43 (No. 1, 2005), 31-45.
%D A000172 J. Franel, Intermediaire des Mathematiciens, 1894.
%D A000172 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM 
               Review, SIAM, 1990; see pp. 148-149.
%D A000172 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               193.
%H A000172 T. D. Noe, <a href="b000172.txt">Table of n, a(n) for n=0..100</a>
%H A000172 Nick Hobson, <a href="a000172.py.txt">Python program for this sequence</
               a>
%H A000172 V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">
               Recurrences and Legendre transform</a>
%H A000172 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinomialSums.html">Link to a section of The World of Mathematics.</
               a>
%H A000172 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FranelNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000172 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SchmidtsProblem.html">Schmidt's Problem</a>
%H A000172 David Callan, <a href="http://front.math.ucdavis.edu/0712.3946">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} </a>.
%F A000172 A002893(n) = Sum_{m=0..n} binomial(n, m) a(m) [Barrucand]
%F A000172 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.
%F A000172 (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
%F A000172 a(n) ~ 2*3^(-1/2)*pi^-1*n^-1*2^(3*n) - Joe Keane (jgk(AT)jgk.org), Jun 
               21 2002
%Y A000172 Cf. A002893, A052144, A005260, A096191. Second row of array A094424.
%Y A000172 Cf. A033581.
%Y A000172 Sequence in context: A122826 A108490 A165817 this_sequence A097971 A093303 
               A075870
%Y A000172 Adjacent sequences: A000169 A000170 A000171 this_sequence A000173 A000174 
               A000175
%K A000172 nonn,easy,nice
%O A000172 0,2
%A A000172 N. J. A. Sloane (njas(AT)research.att.com).