A081798 -id:A081798 - cp's OEIS frontend

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

1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, 38165260, 278415920, 2046924400, 15148345760, 112738423360, 843126957056, 6332299624282, 47737325577620, 361077477684436, 2739270870994736, 20836827035351596, 158883473753259752, 1214171997616258240

Offset: 0

N. J. A. Sloane

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.
This is the Taylor expansion of a special point on a curve described by Beauville. - Matthijs Coster, Apr 28 2004
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
Conjecture: a(n) == 2 (mod n^3) iff n is prime. - Gary Detlefs, Mar 22 2013
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
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
This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
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
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
Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - x*y*z). - Seiichi Manyama, Jul 11 2020
Named after the Swiss mathematician Jérôme Franel (1859-1939). - Amiram Eldar, Jun 15 2021
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

			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 + ... - _Paul D. Hanna_, 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. - _Paul D. Hanna_

Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
Jérôme Franel, On a question of Laisant, Intermédiaire des Mathématiciens, vol 1 1894 pp 45-47
H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 56.
Murray Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 148-149.
John 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).

Indranil Ghosh, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe)
Boris Adamczewski, Jason P. Bell, and Eric Delaygue, Algebraic independence of G-functions and congruences à la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Prarit Agarwal and June Nahmgoong, Singlets in the tensor product of an arbitrary number of Adjoint representations of SU(3), arXiv:2001.10826 [math.RT], 2020.
Richard Askey, Orthogonal Polynomials and Special Functions, SIAM, 1975; see p. 43.
P. Barrucand, A combinatorial identity, Problem 75-4, SIAM Rev., Vol. 17 (1975), p. 168. Solution by D. R. Breach, D. McCarthy, D. Monk and P. E. O'Neil, SIAM Rev., Vol. 18 (1976), p. 303.
P. Barrucand, Problem 75-4, A Combinatorial Identity, SIAM Rev., 17 (1975), 168. [Annotated scanned copy of statement of problem]
Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, Vol.. 294 (May 24 1982), pp. 657-660.
David Callan, 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), arXiv:0712.3946 [math.CO], 2007.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS, Vol. 11 (2008), Article 08.3.4.
Marc Chamberland and Armin Straub, Apéry Limits: Experiments and Proofs, arXiv:2011.03400 [math.NT], 2020.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
M. Coster, Email, Nov 1990
T. W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory, Series A, Vol. 52, No. 1 (1989), pp. 77-83.
Eric Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
Robert W. Donley Jr, Directed path enumeration for semi-magic squares of size three, arXiv:2107.09463 [math.CO], 2021.
Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math., Vol. 308, No. 11 (2008), pp. 2182--2212. MR2404544 (2009j:05019) - From _N. J. A. Sloane_, May 01 2012
Carsten Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., Vol. 43, No. 1 (2005), pp. 31-45.
Jeff D. Farmer and Steven C. Leth, An asymptotic formula for powers of binomial coefficients, Math. Gaz., Vol. 89, No. 516 (2005), pp. 385-391.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See A p. 2.
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See pp. 34-35.
S. Herfurtner, Elliptic surfaces with four singular fibres, Mathematische Annalen, 1991. Preprint.
Nick Hobson, Python program for this sequence.
Lawrence Hollom, Julien Portier, and Victor Souza, Double-jump phase transition for the reverse Littlewood-Offord problem, arXiv:2503.24202 [math.CO], 2025. See p. 7.
Bradley Klee, Checking Weierstrass data, 2023.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 282.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, Vol. 2, No. 5 (2016).
Guo-Shuai Mao, Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers, arXiv:2111.08778 [math.NT], 2021.
Guo-Shuai Mao and Yan Liu, On a congruence conjecture of Z.-W. Sun involving Franel numbers, arXiv:2111.08775 [math.NT], 2021.
Guo-Shuai Mao, On three conjectural congruences involving Domb numbers and Franel numbers, preprint on ResearchGate, April 2024.
Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Marci A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory, Vo. 27 (1987), pp. 304-309.
Juan Pla, Problem H-505, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 33, No. 5 (1995), p. 473; Sum Formulae!, Solution to Problem H-505 by Paul S. Bruckman, ibid., Vol. 35, No. 1 (1997), pp. 93-95.
Armin Straub, and Wadim Zudilin, Sums of powers of binomials, their Apéry limits, and Franel's suspicions, arXiv:2112.09576 [math.NT], 2021.
Volker Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Zhi-Wei Sun, Congruences for Franel numbers, arXiv preprint arXiv:1112.1034 [math.NT], 2011.
Zhi-Wei Sun, Connections between p = x^2+3y^2 and Franel numbers, J. Number Theory, Vol. 133 (2013), pp. 2919-2928.
Zhi-Wei Sun, Conjectures involving arithmetical sequences, 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.
Zhi-Wei Sun, Congruences involving g_n(x) = Sum_{k= 0..n} C(n,k)^2 C(2k,k) x^k, arXiv preprint arXiv:1407.0967 [math.NT], 2014.
Raimundas Vidunas, Counting derangements and Nash equilibria, arxiv 1401.5400 [math.CO], 2014. [Original title: MacMahon's master theorem and totally mixed Nash equilibria]
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.
Jin Yuan, Zhi-Juan Lu, Asmus L. Schmidt, On recurrences for sums of powers of binomial coefficients, J. Numb. Theory 128 (2008) 2784-2794
Don Zagier, Integral solutions of Apéry-like recurrence equations. See line A in sporadic solutions table of page 5.
Bao-Xuan Zhu, Higher order log-monotonicity of combinatorial sequences, arXiv preprint arXiv:1309.6025 [math.CO], 2013.

Cf. A002893, A052144, A005260, A096191, A033581, A189791. Second row of array A094424.
Cf. A181543, A006480, A141057, A000578, A007318.
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.)
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.
Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Column k=3 of A372307.

a000172 = sum . map a000578 . a007318_row
-- Reinhard Zumkeller, Jan 06 2013

A000172 := proc(n)
    add(binomial(n,k)^3,k=0..n) ;
end proc:
seq(A000172(n),n=0..10) ; # R. J. Mathar, Jul 26 2014
A000172_list := proc(len) series(hypergeom([], [1, 1], x)^2, x, len);
seq((n!)^3*coeff(%, x, n), n=0..len-1) end:
A000172_list(21); # Peter Luschny, May 31 2017

Table[Sum[Binomial[n,k]^3,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Aug 24 2011 *)
Table[ HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1], {n, 0, 20}]  (* Jean-François Alcover, Jul 16 2012, after symbolic sum *)
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 *)
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 *)

{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

{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

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

def A000172():
    x, y, n = 1, 2, 1
    while True:
        yield x
        n += 1
        x, y = y, (8*(n-1)^2*x + (7*n^2-7*n + 2)*y) // n^2
a = A000172()
[next(a) for i in range(21)]   # Peter Luschny, Oct 12 2013

A002893(n) = Sum_{m = 0..n} binomial(n, m)*a(m) [Barrucand].
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.
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
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} (3*n)!/n!^3 * x^(2*n)/(1 - 2*x)^(3*n+1). - Paul D. Hanna, Oct 30 2010
G.f.: hypergeom([1/3, 2/3], [1], 27 x^2 / (1 - 2x)^3) / (1 - 2x). - Michael Somos, Dec 17 2010
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
G.f.: A(x) = 1/(1-2*x)*(1+6*(x^2)/(G(0)-6*x^2)),
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
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
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
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
a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^2. - Peter Luschny, May 31 2017
From Gheorghe Coserea, Jul 04 2018: (Start)
a(n) = Sum_{k=0..floor(n/2)} (n+k)!/(k!^3*(n-2*k)!) * 2^(n-2*k).
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)
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
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
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
From Bradley Klee, Jun 05 2023: (Start)
The g.f. T(x) obeys a period-annihilating ODE:
0=2*(1 + 4*x)*T(x) + (-1 + 14*x + 24*x^2)*T'(x) + x*(1 + x)*(-1 + 8*x)*T''(x).
The periods ODE can be derived from the following Weierstrass data:
g2 = (4/243)*(1 - 8*x + 240*x^2 - 464*x^3 + 16*x^4);
g3 = -(8/19683)*(1 - 12*x - 480*x^2 + 3080*x^3 - 12072*x^4 + 4128*x^5 +
    64*x^6);
which determine an elliptic surface with four singular fibers. (End)
From Peter Bala, Oct 31 2024: (Start)
For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^2 * binomial(n-1, k). Cf. A361716.
For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n + 1], [1, 1], -1). (End)

A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 25, 115, 63, 1, 1, 1, 121, 2641, 2371, 321, 1, 1, 1, 721, 114121, 392641, 54091, 1683, 1, 1, 1, 5041, 7489441, 169417921, 67982041, 1307377, 8989, 1, 1, 1, 40321, 681120721, 137322405361, 308238414121, 12838867105, 32803219, 48639, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2013

Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j - Product_{j=1..k} x_j) for k>1. - Seiichi Manyama, Jul 10 2020

			A(1,3) = 3*2+1 = 7:
          (0,1,1)-(0,0,1)
         /       X       \
  (1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
       \ \       X       / /
        \ (1,1,0)-(1,0,0) /
         `---------------´
Square array A(n,k) begins:
  1, 1,    1,       1,           1,               1, ...
  1, 1,    3,       7,          25,             121, ...
  1, 1,   13,     115,        2641,          114121, ...
  1, 1,   63,    2371,      392641,       169417921, ...
  1, 1,  321,   54091,    67982041,    308238414121, ...
  1, 1, 1683, 1307377, 12838867105, 629799991355641, ...

Alois P. Heinz, Antidiagonals n = 0..44, flattened

Columns k=0+1, 2-10 give: A000012, A001850, A081798, A082488, A082489, A229049, A229674, A229675, A229676, A229677.
Rows n=0-1 give: A000012, A038507 (for k>1).
Main diagonal gives: A229267.
Cf. A060854, A227578, A227655, A225094, A210472, A262809, A263159.

with(combinat):
A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

a[, 0] = a[, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.

G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - Seiichi Manyama, Jul 10 2020

A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

Original entry on oeis.org

1, 25, 2641, 392641, 67982041, 12838867105, 2564949195985, 533008982952625, 114035552691160585, 24950692835328410305, 5557138347370070346601, 1255741805437716400557625, 287180884347761929741524361, 66343186345544102086872515761

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1/(1-(x + y + z + w + x*y*z*w)). - Gheorghe Coserea, Jul 15 2016

			G.f.: A(x) = 1 + 25*x + 2641*x^2 + 392641*x^3 + 67982041*x^4 + 12838867105*x^5 +...
where
A(x) = 1/(1-x) + (4!/1!^4)*x/(1-x)^5 + (8!/2!^4)*x^2/(1-x)^9 + (12!/3!^4)*x^3/(1-x)^13 + (16!/4!^4)*x^4/(1-x)^17 + (20!/5!^4)*x^5/(1-x)^21 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 24*x/(1-x)^5 + 2520*x^2/(1-x)^9 + 369600*x^3/(1-x)^13 + 63063000*x^4/(1-x)^17 + 11732745024*x^5/(1-x)^21 +...+ A008977(n)*x^n/(1-x)^(4*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..400
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A081798.
Column k = 4 of A229142.
Cf. A008977, A001850, A082489, A229049, A229050.
Related to diagonal of rational functions: A268545-A268555.

[&+[Binomial(n, k)*Binomial(n+k, k)*Binomial(n+2*k, k)*Binomial(n+3*k, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Oct 16 2018

with(combinat):
a:= n-> add(multinomial(n+3*k, n-k, k$4), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k]* Binomial[n+3*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (4*m)!/m!^4*x^m/(1-x+x*O(x^n))^(4*m+1)), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k)*binomial(n+3*k, k))}
for(n=0, 15, print1(a(n), ", "))

G.f.: Sum_{n>=0} (4*n)!/n!^4 * x^n / (1-x)^(4*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^3*(2*n-3)*(4*n-9)*(4*n-5)*a(n) = (4*n-9)*(4*n-3)*(520*n^4 - 1820*n^3 + 2109*n^2 - 905*n + 121)*a(n-1) - (192*n^6 - 1536*n^5 + 4748*n^4 - 7050*n^3 + 5065*n^2 - 1563*n + 171)*a(n-2) + (4*n-1)*(32*n^5 - 296*n^4 + 1040*n^3 - 1689*n^2 + 1209*n - 279)*a(n-3) - (n-3)^3*(2*n-1)*(4*n-5)*(4*n-1)*a(n-4). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(Pi^(3/2)*n^(3/2)), where d = 65 + 46*sqrt(2) + 2*sqrt(2*(1055 + 746*sqrt(2))) = 259.976980158726979... is the maximal positive root of the equation 1 - 4*d + 6*d^2 - 260*d^3 + d^4 = 0 and c = sqrt(8 + 5*sqrt(2) + sqrt(14*(11 + 8*sqrt(2))))/8 = 0.71529801573844067904424114047445568721... - Vaclav Kotesovec, Sep 23 2013, updated Jul 16 2016
G.f.: hypergeom([1/8, 3/8],[1],256*x/(1-x)^4)^2/(1-x). - Mark van Hoeij, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 13*x^2 + 893*x^3 + 99125*x^4 + 13706093*x^5 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016
0 = x^2*(3*x+1)^2*(1-260*x+6*x^2-4*x^3+x^4)*y''' + 3*x*(3*x+1)*(1-390*x-378*x^2+8*x^3-15*x^4+6*x^5)*y'' + (1-836*x+133*x^2+768*x^3-69*x^4-60*x^5+63*x^6)*y' + (-25+397*x-378*x^2-6*x^3+3*x^4+9*x^5)*y, where y is the g.f. - Gheorghe Coserea, Jul 15 2016

A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

Original entry on oeis.org

1, 121, 114121, 169417921, 308238414121, 629799991355641, 1387152264043496161, 3220175519103433952161, 7771784978946238318454761, 19326687177288750280293146161, 49215884415076728067274047737961, 127771596843320597524806463425540481

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1 / (1 - x - y - z - u - v - x*y*z*u*v). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 121*x + 114121*x^2 + 169417921*x^3 + 308238414121*x^4 +...
where
A(x) = 1/(1-x) + (5!/1!^5)*x/(1-x)^6 + (10!/2!^5)*x^2/(1-x)^11 + (15!/3!^5)*x^3/(1-x)^16 + (20!/4!^5)*x^4/(1-x)^21 + (25!/5!^5)*x^5/(1-x)^26 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 120*x/(1-x)^6 + 113400*x^2/(1-x)^11 + 168168000*x^3/(1-x)^16 + 305540235000*x^4/(1-x)^21 + 623360743125120*x^5/(1-x)^26 +...+ A008978(n)*x^n/(1-x)^(5*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k = 5 of A229142.

Cf. A008978, A001850, A081798, A082488, A229049, A229050.

with(combinat):
a:= n-> add(multinomial(n+4*k, n-k, k$5), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k]* Binomial[n+3*k,k]*Binomial[n+4*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (5*m)!/m!^5*x^m/(1-x+x*O(x^n))^(5*m+1)), n)}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 22 2013

{a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k) *binomial(n+3*k, k)*binomial(n+4*k, k))}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 22 2013

G.f.: Sum_{n>=0} (5*n)!/n!^5 * x^n / (1-x)^(5*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^4*(5*n-16)*(5*n-12)*(5*n-11)*(5*n-8)*(5*n-7)*(5*n-6)*a(n) = (5*n-16)*(5*n-12)*(5*n-11)*(5*n-4)*(78250*n^6 - 422550*n^5 + 885665*n^4 - 906704*n^3 + 468906*n^2 - 114379*n + 10086)*a(n-1) - (5*n-16)*(31250*n^9 - 400000*n^8 + 2154375*n^7 - 6337750*n^6 + 11073100*n^5 - 11721380*n^4 + 7379043*n^3 - 2629646*n^2 + 489456*n - 36000)*a(n-2) + (5*n-1)*(31250*n^9 - 556250*n^8 + 4241875*n^7 - 18056500*n^6 + 46858025*n^5 - 76033760*n^4 + 76116292*n^3 - 44628880*n^2 + 13702848*n - 1693440)*a(n-3) - (5*n-6)*(5*n-2)*(5*n-1)*(625*n^7 - 11375*n^6 + 86025*n^5 - 347305*n^4 + 798274*n^3 - 1025292*n^2 + 661408*n - 156480)*a(n-4) + (n-4)^4*(5*n-11)*(5*n-7)*(5*n-6)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-5). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/n^2, where d = 3129.996806129131084... is the root of the equation -1 + 5*d - 10*d^2 + 10*d^3 - 3130*d^4 +d^5 = 0 and c = 0.05674890286773483081841276583916042181... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 61*x^2 + 38101*x^3 + 42394381*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

Original entry on oeis.org

1, 1, 7, 25, 151, 751, 4411, 24697, 146455, 862351, 5195257, 31392967, 191815339, 1177508515, 7276161907, 45154764025, 281492498455, 1761076827895, 11055132835705, 69600761349175, 439370198255401, 2780265190892641, 17631718101804517, 112038660509078695

Offset: 0

Paul D. Hanna, Feb 26 2012

nonn

Compare g.f. to: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1), which is a g.f. of the Franel numbers (A000172).
From Zhi-Wei Sun, Nov 12 2016: (Start)
Conjecture: (i) For any prime p > 3 and positive integer n, the number (a(p*n)-a(n))/(p*n)^3 is always a p-adic integer.
(ii) For any prime p == 1 (mod 3), we have Sum_{k=0..p-1}a(k) == C(2(p-1)/3,(p-1)/3) (mod p^2). For any prime p == 2 (mod 3), we have Sum_{k=0..p-1}a(k) == 2p/C(2(p+1)/3,(p+1)/3) (mod p^2).
We have proved part (i) of this conjecture for n = 1. (End)
Diagonal of rational functions 1/(1 - x*y - y*z - x*z - x*y*z), 1/(1 - x*y + y*z + x*z - x*y*z). - Gheorghe Coserea, Jul 03 2018
Number of paths from (0,0,0) to (n,n,n) using steps (1,1,0), (1,0,1), (0,1,1), and (1,1,1). - William J. Wang, Dec 07 2020
Diagonal of the rational function 1/(1 - (x^2 + y^2 + z^2 + x*y*z)). - Seiichi Manyama, Jul 04 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 25*x^3 + 151*x^4 + 751*x^5 + 4411*x^6 +...
where
A(x) = 1/(1-x) + 6*x^2/(1-x)^4 + 90*x^4/(1-x)^7 + 1680*x^6/(1-x)^10 + 34650*x^8/(1-x)^13 + 756756*x^10/(1-x)^16 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Hao Pan and Zhi-Wei Sun, Supercongruences for central trinomial coefficients, arXiv:2012.05121 [math.NT], 2020.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000172, A001850, A208426, A244973, A274783.

Cf. A081798, A344560.

series(hypergeom([1/3, 2/3], [1], 27*x^2/(1 - x)^3)/(1 - x), x=0, 25): seq(coeff(%, x, n), n=0..23);  # Mark van Hoeij, May 20 2013
a := n -> hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4); seq(simplify(a(n)), n=0..23);  # Peter Luschny, Jan 11 2025

nmax = 20; CoefficientList[Series[Sum[(3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1), {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)/(1-x+x*O(x^n))^(3*m+1)),n)}
for(n=0,25,print1(a(n),", "))

Conjecture: n^2*(3*n-5)*a(n) +(-9*n^3+24*n^2-17*n+4) *a(n-1) -(3*n-4) *(24*n^2-56*n+27)*a(n-2) -(3*n-2)*(n-2)^2*a(n-3)=0. - R. J. Mathar, Mar 10 2016
a(n) ~ sqrt(1/2 + sqrt(13)*cos(arctan(53*sqrt(3)/19)/3)/6) * (1 + 6*cos(Pi/9))^n / (Pi*n). - Vaclav Kotesovec, Jul 05 2016
It is easy to show that a(n) = Sum_{k=0..n}C(n,k)*C(n-k,k)*C(n+k,k) = Sum_{k=0..n}C(n+k,k)*C(n,2k)*C(2k,k). By this formula and the Zeilberger algorithm, we confirm the recurrence conjectured by R. J. Mathar. - Zhi-Wei Sun, Nov 12 2016
G.f. y=A(x) satisfies: 0 = x*(x + 2)*(x^3 + 24*x^2 + 3*x - 1)*y'' + (3*x^4 + 56*x^3 + 147*x^2 + 12*x - 2)*y' + (x^3 + 9*x^2 + 42*x + 2)*y. - Gheorghe Coserea, Jul 03 2018
a(n) = hypergeom([1/2 - n/2, -n/2, n + 1], [1, 1], 4). - Peter Luschny, Jan 11 2025

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).

Original entry on oeis.org

1, 721, 7489441, 137322405361, 3249278494922881, 88913838899388373921, 2672932604015450235911761, 85821373873727899097881489201, 2892941442791065880984595547275841, 101239016762863657789924022384195015281, 3649717311362112161867915447690005522733041

Offset: 0

Paul D. Hanna, Sep 22 2013

Diagonal of the rational function 1 / (1 - x - y - z - u - v - w - x*y*z*u*v*w). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 721*x + 7489441*x^2 + 137322405361*x^3 + 3249278494922881*x^4 +...
where
A(x) = 1/(1-x) + (6!/1!^6)*x/(1-x)^7 + (12!/2!^6)*x^2/(1-x)^13 + (18!/3!^6)*x^3/(1-x)^19 + (24!/4!^6)*x^4/(1-x)^25 + (30!/5!^6)*x^5/(1-x)^31 +...
Equivalently,
A(x) = 1/(1-x) + 720*x/(1-x)^7 + 7484400*x^2/(1-x)^13 + 137225088000*x^3/(1-x)^19 + 3246670537110000*x^4/(1-x)^25 + 88832646059788350720*x^5/(1-x)^31 +...+ A008979(n)*x^n/(1-x)^(6*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A008979, A001850, A081798, A082488, A082489, A229050.

Column k = 6 of A229142.

with(combinat):
a:= n-> add(multinomial(n+5*k, n-k, k$6), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k] *Binomial[n+3*k,k] *Binomial[n+4*k,k]*Binomial[n+5*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (6*m)!/m!^6*x^m/(1-x+x*O(x^n))^(6*m+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k) *binomial(n+2*k,k) *binomial(n+3*k,k) *binomial(n+4*k,k)*binomial(n+5*k,k))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k) * C(n+5*k,k).
Recurrence: n^5*(2*n - 5)*(2*n - 3)*(3*n - 10)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 7)*a(n) = (2*n - 5)*(3*n - 10)*(3*n - 7)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 5)*(839916*n^8 - 6159384*n^7 + 18804591*n^6 - 30967129*n^5 + 29803190*n^4 - 16984623*n^3 + 5534242*n^2 - 929843*n + 60482)*a(n-1) - (3*n - 10)*(6*n - 25)*(6*n - 19)*(58320*n^12 - 1030320*n^11 + 7973640*n^10 - 35550360*n^9 + 101096973*n^8 - 191892891*n^7 + 247426961*n^6 - 216687345*n^5 + 127127767*n^4 - 48662719*n^3 + 11593839*n^2 - 1535715*n + 84350)*a(n-2) + 10*(6*n - 25)*(6*n - 1)*(23328*n^13 - 618192*n^12 + 7344432*n^11 - 51616440*n^10 + 238504338*n^9 - 761904909*n^8 + 1722993100*n^7 - 2778206390*n^6 + 3175831572*n^5 - 2526793076*n^4 + 1352618106*n^3 - 459806772*n^2 + 89082095*n - 7435050)*a(n-3) - 5*(3*n - 1)*(6*n - 7)*(6*n - 1)*(11664*n^12 - 369360*n^11 + 5235192*n^10 - 43777800*n^9 + 239670873*n^8 - 901183065*n^7 + 2374616540*n^6 - 4392523494*n^5 + 5622136222*n^4 - 4816276070*n^3 + 2596763070*n^2 - 784074950*n + 100205500)*a(n-4) + (2*n - 1)*(3*n - 4)*(3*n - 1)*(6*n - 13)*(6*n - 7)*(6*n - 1)*(648*n^9 - 19548*n^8 + 256338*n^7 - 1909293*n^6 + 8851093*n^5 - 26285080*n^4 + 49492875*n^3 - 56141750*n^2 + 34024625*n - 8063750)*a(n-5) - (n-5)^5*(2*n - 3)*(2*n - 1)*(3*n - 7)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(6*n - 19)*(6*n - 13)*(6*n - 7)*(6*n - 1)*a(n-6). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(n^(5/2)), where d = 46661.9996785484656481246... is the root of the equation 1 - 6*d + 15*d^2 - 20*d^3 + 15*d^4 - 46662*d^5 + d^6 = 0 and c = 0.024758197509539176365175770882978221... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 361*x^2 + 2496841*x^3 + 34333162981*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

Diagonal of rational function R(x,y,z,t) = 1/(1 - (x + y + z + t*x*y*z)) with respect to x,y,z, i.e., T(n,k) = [(xyz)^n*t^k] R(x,y,z,t).

Annihilating differential operator: x*(2*t*x + 1)*((t*x - 1)^3 + 27*x)*Dx^2 + (6*t^4*x^4 - 8*t^3*x^3 - 3*t*(t - 18)*x^2 + 6*(t + 9)*x - 1)*Dx + (t*x - 1)*(t*(2*t^2*x^2 + 2*t*x - 1) - 6).

cp's OEIS Frontend

A082487 Duplicate of A081798.

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

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A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k).

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A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k).

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A208425 Expansion of Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-x)^(3*n+1).

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A229049 G.f.: Sum_{n >= 0} (6*n)!/n!^6 * x^n / (1-x)^(6*n+1).

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

A208425 Expansion of Sum_{n>=0} (3n)!/n!^3 x^(2n)/(1-x)^(3n+1).

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).

A318107 Triangle read by rows: T(n,k) = (3n - 2k)!/((n-k)!^3*k!).