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 (jvospost2(AT)yahoo.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.

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Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.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

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

Nick Hobson, Python program for this sequence

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.

Sequence in context: A102536 A122826 A108490 this_sequence A097971 A093303 A075870

Adjacent sequences: A000169 A000170 A000171 this_sequence A000173 A000174 A000175

nonn,easy,nice

njas