0,2

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

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

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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

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

Nick Hobson, Python program for this sequence

V. Strehl, Recurrences and Legendre transform

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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} .

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

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

Cf. A033581.

Adjacent sequences: A000169 A000170 A000171 this_sequence A000173 A000174 A000175

Sequence in context: A152395 A122826 A108490 this_sequence A097971 A093303 A075870

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).