cp's OEIS Frontend

A000172 The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.

%I A000172 M1971 N0781 #381 Aug 21 2025 11:16:41
%S A000172 1,2,10,56,346,2252,15184,104960,739162,5280932,38165260,278415920,
%T A000172 2046924400,15148345760,112738423360,843126957056,6332299624282,
%U A000172 47737325577620,361077477684436,2739270870994736,20836827035351596,158883473753259752,1214171997616258240
%N A000172 The Franel number a(n) = Sum_{k = 0..n} binomial(n,k)^3.
%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 floor((r+3)/2) terms.
%C A000172 This is the Taylor expansion of a special point on a curve described by Beauville. - _Matthijs Coster_, Apr 28 2004
%C A000172 An identity of V. Strehl states that a(n) = Sum_{k = 0..n} C(n,k)^2 * binomial(2*k,n). Zhi-Wei Sun conjectured that for every n = 2,3,... the polynomial f_n(x) = Sum_{k = 0..n} binomial(n,k)^2 * binomial(2*k,n) * x^(n-k) is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 21 2013
%C A000172 Conjecture: a(n) == 2 (mod n^3) iff n is prime. - _Gary Detlefs_, Mar 22 2013
%C A000172 a(p) == 2 (mod p^3) for any prime p since p | C(p,k) for all k = 1,...,p-1. - _Zhi-Wei Sun_, Aug 14 2013
%C A000172 a(n) is the maximal number of totally mixed Nash equilibria in games of 3 players, each with n+1 pure options. - _Raimundas Vidunas_, Jan 22 2014
%C A000172 This is one of the Apéry-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A000172 Diagonal of rational functions 1/(1 - x*y - y*z - x*z - 2*x*y*z), 1/(1 - x - y - z + 4*x*y*z), 1/(1 + y + z + x*y + y*z + x*z + 2*x*y*z), 1/(1 + x + y + z + 2*(x*y + y*z + x*z) + 4*x*y*z). - _Gheorghe Coserea_, Jul 04 2018
%C A000172 a(n) is the constant term in the expansion of ((1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - _Seiichi Manyama_, Oct 27 2019
%C A000172 Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - x*y*z). - _Seiichi Manyama_, Jul 11 2020
%C A000172 Named after the Swiss mathematician Jérôme Franel (1859-1939). - _Amiram Eldar_, Jun 15 2021
%C A000172 It appears that a(n) is equal to the coefficient of (x*y*z)^n in the expansion of (1 + x + y - z)^n * (1 + x - y + z)^n * (1 - x + y + z)^n. Cf. A036917. - _Peter Bala_, Sep 20 2021
%D A000172 Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
%D A000172 Jérôme Franel, On a question of Laisant, Intermédiaire des Mathématiciens, vol 1 1894 pp 45-47
%D A000172 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 56.
%D A000172 Murray Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
%D A000172 John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.
%D A000172 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000172 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000172 Indranil Ghosh, <a href="/A000172/b000172.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from T. D. Noe)
%H A000172 Boris Adamczewski, Jason P. Bell, and Eric Delaygue, <a href="https://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences à la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.
%H A000172 Prarit Agarwal and June Nahmgoong, <a href="https://arxiv.org/abs/2001.10826">Singlets in the tensor product of an arbitrary number of Adjoint representations of SU(3)</a>, arXiv:2001.10826 [math.RT], 2020.
%H A000172 Richard Askey, <a href="https://dx.doi.org/10.1137/1.9781611970470">Orthogonal Polynomials and Special Functions</a>, SIAM, 1975; see p. 43.
%H A000172 P. Barrucand, <a href="https://dx.doi.org/10.1137/1017013">A combinatorial identity, Problem 75-4</a>, SIAM Rev., Vol. 17 (1975), p. 168. <a href="https://dx.doi.org/10.1137/1018056">Solution</a> by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev., Vol. 18 (1976), p. 303.
%H A000172 P. Barrucand, <a href="/A002893/a002893.pdf">Problem 75-4, A Combinatorial Identity</a>, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]
%H A000172 Arnaud Beauville, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k5543443c/f31.item">Les familles stables de courbes sur P_1 admettant quatre fibres singulières</a>, Comptes Rendus, Académie Sciences Paris, Vol.. 294 (May 24 1982), pp. 657-660.
%H A000172 David Callan, <a href="https://arxiv.org/abs/0712.3946">A combinatorial interpretation for the identity Sum_{k=0..n} binomial(n,k) Sum_{j=0..k} binomial(k,j)^3= Sum_{k=0..n} binomial(n,k)^2 binomial(2k,k)</a>, arXiv:0712.3946 [math.CO], 2007.
%H A000172 David Callan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Callan2/callan204.html">A combinatorial interpretation for an identity of Barrucand</a>, JIS, Vol. 11 (2008), Article 08.3.4.
%H A000172 Marc Chamberland and Armin Straub, <a href="https://arxiv.org/abs/2011.03400">Apéry Limits: Experiments and Proofs</a>, arXiv:2011.03400 [math.NT], 2020.
%H A000172 Shaun Cooper, <a href="https://arxiv.org/abs/2302.00757">Apéry-like sequences defined by four-term recurrence relations</a>, arXiv:2302.00757 [math.NT], 2023.
%H A000172 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A000172 T. W. Cusick, <a href="https://dx.doi.org/10.1016/0097-3165(89)90063-0">Recurrences for sums of powers of binomial coefficients</a>, J. Combin. Theory, Series A, Vol. 52, No. 1 (1989), pp. 77-83.
%H A000172 Eric Delaygue, <a href="https://arxiv.org/abs/1310.4131">Arithmetic properties of Apéry-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
%H A000172 Robert W. Donley Jr, <a href="https://arxiv.org/abs/2107.09463">Directed path enumeration for semi-magic squares of size three</a>, arXiv:2107.09463 [math.CO], 2021.
%H A000172 Tomislav Došlic and Darko Veljan, <a href="https://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math., Vol. 308, No. 11 (2008), pp. 2182--2212. MR2404544 (2009j:05019) - From _N. J. A. Sloane_, May 01 2012
%H A000172 Carsten Elsner, <a href="https://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulae for sums involving binomial coefficients</a>, Fib. Q., Vol. 43, No. 1 (2005), pp. 31-45.
%H A000172 Jeff D. Farmer and Steven C. Leth, <a href="https://www.jstor.org/stable/3621929">An asymptotic formula for powers of binomial coefficients</a>, Math. Gaz., Vol. 89, No. 516 (2005), pp. 385-391.
%H A000172 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See A p. 2.
%H A000172 Darij Grinberg, <a href="https://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf">Introduction to Modern Algebra</a> (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
%H A000172 S. Hassani, J.-M. Maillard, and N. Zenine, <a href="https://arxiv.org/abs/2502.05543">On the diagonals of rational functions: the minimal number of variables (unabridged version)</a>, arXiv:2502.05543 [math-ph], 2025. See pp. 34-35.
%H A000172 S. Herfurtner, <a href="https://doi.org/10.1007/BF01445211">Elliptic surfaces with four singular fibres</a>, Mathematische Annalen, 1991. <a href="https://archive.mpim-bonn.mpg.de/id/eprint/860/">Preprint</a>.
%H A000172 Nick Hobson, <a href="/A000172/a000172.py.txt">Python program for this sequence</a>.
%H A000172 Lawrence Hollom, Julien Portier, and Victor Souza, <a href="https://arxiv.org/abs/2503.24202">Double-jump phase transition for the reverse Littlewood-Offord problem</a>, arXiv:2503.24202 [math.CO], 2025. See p. 7.
%H A000172 Bradley Klee, <a href="/A006077/a006077.pdf">Checking Weierstrass data</a>, 2023.
%H A000172 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 282.
%H A000172 Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, Vol. 2, No. 5 (2016).
%H A000172 Guo-Shuai Mao, <a href="https://arxiv.org/abs/2111.08778">Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers</a>, arXiv:2111.08778 [math.NT], 2021.
%H A000172 Guo-Shuai Mao and Yan Liu, <a href="https://arxiv.org/abs/2111.08775">On a congruence conjecture of Z.-W. Sun involving Franel numbers</a>, arXiv:2111.08775 [math.NT], 2021.
%H A000172 Guo-Shuai Mao, <a href="https://dx.doi.org/10.13140/RG.2.2.28230.48964">On three conjectural congruences involving Domb numbers and Franel numbers</a>, preprint on ResearchGate, April 2024.
%H A000172 Romeo Meštrović, <a href="https://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H A000172 Marci A. Perlstadt, <a href="https://dx.doi.org/10.1016/0022-314X(87)90069-2">Some Recurrences for Sums of Powers of Binomial Coefficients</a>, Journal of Number Theory, Vo. 27 (1987), pp. 304-309.
%H A000172 Juan Pla, <a href="https://www.fq.math.ca/Scanned/33-5/advanced33-5.pdf">Problem H-505</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 33, No. 5 (1995), p. 473; <a href="https://www.fq.math.ca/Scanned/35-1/advanced35-1.pdf">Sum Formulae!</a>, Solution to Problem H-505 by Paul S. Bruckman, ibid., Vol. 35, No. 1 (1997), pp. 93-95.
%H A000172 Armin Straub, and Wadim Zudilin, <a href="https://arxiv.org/abs/2112.09576">Sums of powers of binomials, their Apéry limits, and Franel's suspicions</a>, arXiv:2112.09576 [math.NT], 2021.
%H A000172 Volker Strehl, <a href="https://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
%H A000172 Zhi-Hong Sun, <a href="https://arxiv.org/abs/1803.10051">Congruences for Apéry-like numbers</a>, arXiv:1803.10051 [math.NT], 2018.
%H A000172 Zhi-Hong Sun, <a href="https://arxiv.org/abs/2004.07172">New congruences involving Apéry-like numbers</a>, arXiv:2004.07172 [math.NT], 2020.
%H A000172 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1112.1034">Congruences for Franel numbers</a>, arXiv preprint arXiv:1112.1034 [math.NT], 2011.
%H A000172 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/150f.pdf">Connections between p = x^2+3y^2 and Franel numbers</a>, J. Number Theory, Vol. 133 (2013), pp. 2919-2928.
%H A000172 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1208.2683">Conjectures involving arithmetical sequences</a>, arXiv:1208.2683v9 [math.CO] 2013; 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.
%H A000172 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1407.0967">Congruences involving g_n(x) = Sum_{k= 0..n} C(n,k)^2 C(2k,k) x^k</a>, arXiv preprint arXiv:1407.0967 [math.NT], 2014.
%H A000172 Raimundas Vidunas, <a href="https://arxiv.org/abs/1401.5400">Counting derangements and Nash equilibria</a>, arxiv 1401.5400 [math.CO], 2014. [Original title: MacMahon's master theorem and totally mixed Nash equilibria]
%H A000172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>.
%H A000172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FranelNumber.html">Franel Number</a>.
%H A000172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.
%H A000172 Jin Yuan, Zhi-Juan Lu, Asmus L. Schmidt, <a href="https://dx.doi.org/10.1016/j.jnt.2008.03.011">On recurrences for sums of powers of binomial coefficients</a>, J. Numb. Theory 128 (2008) 2784-2794
%H A000172 Don Zagier, <a href="https://people.mpim-bonn.mpg.de/zagier/files/tex/AperylikeRecEqs/fulltext.pdf">Integral solutions of Apéry-like recurrence equations</a>. See line A in sporadic solutions table of page 5.
%H A000172 Bao-Xuan Zhu, <a href="https://arxiv.org/abs/1309.6025">Higher order log-monotonicity of combinatorial sequences</a>, arXiv preprint arXiv:1309.6025 [math.CO], 2013.
%F A000172 A002893(n) = Sum_{m = 0..n} binomial(n, m)*a(m) [Barrucand].
%F A000172 Sum_{k = 0..n} C(n, k)^3 = (-1)^n*Integral_{x = 0..infinity} L_k(x)^3 exp(-x) dx. - from Askey's book, p. 43.
%F A000172 D-finite with recurrence (n + 1)^2*a(n+1) = (7*n^2 + 7*n + 2)*a(n) + 8*n^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
%F A000172 O.g.f.: A(x) = Sum_{n >= 0} (3*n)!/n!^3 * x^(2*n)/(1 - 2*x)^(3*n+1). - _Paul D. Hanna_, Oct 30 2010
%F A000172 G.f.: hypergeom([1/3, 2/3], [1], 27 x^2 / (1 - 2x)^3) / (1 - 2x). - _Michael Somos_, Dec 17 2010
%F A000172 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
%F A000172 G.f.: A(x) = 1/(1-2*x)*(1+6*(x^2)/(G(0)-6*x^2)),
%F A000172 with 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
%F A000172 In 2011 _Zhi-Wei Sun_ found the formula Sum_{k = 0..n} C(2*k,n)*C(2*k,k)*C(2*(n-k),n-k) = (2^n)*a(n) and proved it via the Zeilberger algorithm. - _Zhi-Wei Sun_, Mar 20 2013
%F A000172 0 = a(n)*(a(n+1)*(-2048*a(n+2) - 3392*a(n+3) + 768*a(n+4)) + a(n+2)*(-1280*a(n+2) - 2912*a(n+3) + 744*a(n+4)) + a(n+3)*(+288*a(n+3) - 96*a(n+4))) + a(n+1)*(a(n+1)*(-704*a(n+2) - 1232*a(n+3) + 288*a(n+4)) + a(n+2)*(-560*a(n+2) - 1372*a(n+3) + 364*a(n+4)) + a(n+3)*(+154*a(n+3) - 53*a(n+4))) + a(n+2)*(a(n+2)*(+24*a(n+2) + 70*a(n+3) - 20*a(n+4)) + a(n+3)*(-11*a(n+3) + 4*a(n+4))) for all n in Z. - _Michael Somos_, Jul 16 2014
%F A000172 For r a nonnegative integer, Sum_{k = r..n} C(k,r)^3*C(n,k)^3 = C(n,r)^3*a(n-r), where we take a(n) = 0 for n < 0. - _Peter Bala_, Jul 27 2016
%F A000172 a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^2. - _Peter Luschny_, May 31 2017
%F A000172 From _Gheorghe Coserea_, Jul 04 2018: (Start)
%F A000172 a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 2^(n-2*k).
%F A000172 G.f. y=A(x) satisfies: 0 = x*(x + 1)*(8*x - 1)*y'' + (24*x^2 + 14*x - 1)*y' + 2*(4*x + 1)*y. (End)
%F A000172 a(n) = [x^n] (1 - x^2)^n*P(n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 56. - _Peter Bala_, Mar 24 2022
%F A000172 a(n) = (2^n/(4*Pi^2)) * Integral_{x,y=0..2*Pi} (1+cos(x)+cos(y)+cos(x+y))^n dx dy = (8^n/(Pi^2)) * Integral_{x,y=0..Pi} (cos(x)*cos(y)*cos(x+y))^n dx dy (Pla, 1995). - _Amiram Eldar_, Jul 16 2022
%F A000172 a(n) = Sum_{k = 0..n} m^(n-k)*binomial(n,k)*binomial(n+2*k,n)*binomial(2*k,k) at m = -4. Cf. A081798 (m = 1), A006480 (m = 0), A124435 (m = -1), A318109 (m = -2) and A318108 (m = -3). - _Peter Bala_, Mar 16 2023
%F A000172 From _Bradley Klee_, Jun 05 2023: (Start)
%F A000172 The g.f. T(x) obeys a period-annihilating ODE:
%F A000172 0=2*(1 + 4*x)*T(x) + (-1 + 14*x + 24*x^2)*T'(x) + x*(1 + x)*(-1 + 8*x)*T''(x).
%F A000172 The periods ODE can be derived from the following Weierstrass data:
%F A000172 g2 = (4/243)*(1 - 8*x + 240*x^2 - 464*x^3 + 16*x^4);
%F A000172 g3 = -(8/19683)*(1 - 12*x - 480*x^2 + 3080*x^3 - 12072*x^4 + 4128*x^5 +
%F A000172     64*x^6);
%F A000172 which determine an elliptic surface with four singular fibers. (End)
%F A000172 From _Peter Bala_, Oct 31 2024: (Start)
%F A000172 For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). Cf. A361716.
%F A000172 For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n + 1], [1, 1], -1). (End)
%e A000172 O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 56*x^3 + 346*x^4 + 2252*x^5 + ...
%e A000172 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 + ... - _Paul D. Hanna_, Oct 30 2010
%e A000172 Let g.f. A(x) = Sum_{n >= 0} a(n)*x^n/n!^3, then
%e A000172 A(x) = 1 + 2*x + 10*x^2/2!^3 + 56*x^3/3!^3 + 346*x^4/4!^3 + ... where
%e A000172 A(x) = [1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 + ...]^2. - _Paul D. Hanna_
%p A000172 A000172 := proc(n)
%p A000172     add(binomial(n,k)^3,k=0..n) ;
%p A000172 end proc:
%p A000172 seq(A000172(n),n=0..10) ; # _R. J. Mathar_, Jul 26 2014
%p A000172 A000172_list := proc(len) series(hypergeom([], [1, 1], x)^2, x, len);
%p A000172 seq((n!)^3*coeff(%, x, n), n=0..len-1) end:
%p A000172 A000172_list(21); # _Peter Luschny_, May 31 2017
%t A000172 Table[Sum[Binomial[n,k]^3,{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Aug 24 2011 *)
%t A000172 Table[ HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1], {n, 0, 20}]  (* _Jean-François Alcover_, Jul 16 2012, after symbolic sum *)
%t A000172 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_ *)
%t A000172 a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/3, 2/3, 1, 27 x^2 / (1 - 2 x)^3] / (1 - 2 x), {x, 0, n}]; (* _Michael Somos_, Jul 16 2014 *)
%o A000172 (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
%o A000172 (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
%o A000172 (PARI) A000172(n)={sum(k=0,(n-1)\2,binomial(n,k)^3)*2+if(!bittest(n,0),binomial(n,n\2)^3)} \\ _M. F. Hasler_, Sep 21 2015
%o A000172 (Haskell)
%o A000172 a000172 = sum . map a000578 . a007318_row
%o A000172 -- _Reinhard Zumkeller_, Jan 06 2013
%o A000172 (Sage)
%o A000172 def A000172():
%o A000172     x, y, n = 1, 2, 1
%o A000172     while True:
%o A000172         yield x
%o A000172         n += 1
%o A000172         x, y = y, (8*(n-1)^2*x + (7*n^2-7*n + 2)*y) // n^2
%o A000172 a = A000172()
%o A000172 [next(a) for i in range(21)]   # _Peter Luschny_, Oct 12 2013
%Y A000172 Cf. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.
%Y A000172 Cf. A181543, A006480, A141057, A000578, A007318.
%Y A000172 The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%Y A000172 For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
%Y A000172 Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
%Y A000172 Column k=3 of A372307.
%K A000172 nonn,easy,nice,changed
%O A000172 0,2
%A A000172 _N. J. A. Sloane_