A181543 -id:A181543 - cp's OEIS frontend

A216207 Matrix inverse of A181543 (cubes of entries of Pascal's triangle).

1, -1, 1, 7, -8, 1, -163, 189, -27, 1, 8983, -10432, 1512, -64, 1, -966751, 1122875, -163000, 7000, -125, 1, 179781181, -208818216, 30317625, -1304000, 23625, -216, 1, -53090086057, 61664945083, -8953081011, 385146125, -6988625, 64827, -343, 1, 23402291822743, -27182124061184, 3946556485312, -169776943616, 3081169000, -28625408, 153664, -512, 1

Offset: 0

R. J. Mathar, Mar 12 2013

Is the leftmost column derived from A212856?

			1;
-1,1;
7,-8,1;
-163,189,-27,1;
8983,-10432,1512,-64,1;
-966751,1122875,-163000,7000,-125,1;
179781181,-208818216,30317625,-1304000,23625,-216,1;

Cf. A055133.

Sum_{j=k..n} A181543(n,j)*T(j,k) = delta(n,k).

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

Original entry on oeis.org

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)

A008459 Square the entries of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 36, 16, 1, 1, 25, 100, 100, 25, 1, 1, 36, 225, 400, 225, 36, 1, 1, 49, 441, 1225, 1225, 441, 49, 1, 1, 64, 784, 3136, 4900, 3136, 784, 64, 1, 1, 81, 1296, 7056, 15876, 15876, 7056, 1296, 81, 1, 1, 100, 2025, 14400, 44100, 63504, 44100, 14400, 2025, 100, 1

Offset: 0

N. J. A. Sloane

Number of lattice paths from (0, 0) to (n, n) with steps (1, 0) and (0, 1), having k right turns. - Emeric Deutsch, Nov 23 2003
Product of A007318 and A105868. - Paul Barry, Nov 15 2005
Number of partitions that fit in an n X n box with Durfee square k. - Franklin T. Adams-Watters, Feb 20 2006
From Peter Bala, Oct 23 2008: (Start)
Narayana numbers of type B. Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p. 60]. See A063007 for the corresponding f-vectors for associahedra of type B_n. See A001263 for the h-vectors for associahedra of type A_n. The Hilbert transform of this triangular array is A108625 (see A145905 for the definition of this term).
Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i, j <= n + 1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the h-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A063007 is the corresponding array of f-vectors for these type A_n polytopes. See A086645 for the array of h-vectors for type C_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
(End)
The n-th row consists of the coefficients of the polynomial P_n(t) = Integral_{s = 0..2*Pi} (1 + t^2 - 2*t*cos(s))^n/Pi/2 ds. For example, when n = 3, we get P_3(t) = t^6 + 9*t^4 + 9*t^2 + 1; the coefficients are 1, 9, 9, 1. - Theodore Kolokolnikov, Oct 26 2010
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!^2 as defined in Wang and Wang. - Peter Bala, Jul 24 2013
From Colin Defant, Sep 16 2018: (Start)
Let s denote West's stack-sorting map. T(n,k) is the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 231, and 321. T(n,k) is also the number of permutations pi of [n+1] with k descents such that s(pi) avoids the patterns 132, 312, and 321.
T(n,k) is the number of permutations of [n+1] with k descents that avoid the patterns 1342, 3142, 3412, and 3421. (End)
The number of convex polyominoes whose smallest bounding rectangle has size (k+1)*(n+1-k) and which contain the lower left corner of the bounding rectangle (directed convex polyominoes). - Günter Rote, Feb 27 2019
Let P be the poset [n] X [n] ordered by the product order.  T(n,k) is the number of antichains in P containing exactly k elements. Cf. A063746. - Geoffrey Critzer, Mar 28 2020

			Pascal's triangle begins
  1
  1  1
  1  2   1
  1  3   3   1
  1  4   6   4   1
  1  5  10  10   5   1
  1  6  15  20  15   6   1
  1  7  21  35  35  21   7   1
...
so the present triangle begins
  1
  1   1
  1   4    1
  1   9    9     1
  1  16   36    16     1
  1  25  100   100    25    1
  1  36  225   400   225   36   1
  1  49  441  1225  1225  441  49   1
...

T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 12.
J. Riordan, An introduction to combinatorial analysis, Dover Publications, Mineola, NY, 2002, page 191, Problem 15. MR1949650
P. G. Tait, On the Linear Differential Equation of the Second Order, Proceedings of the Royal Society of Edinburgh, 9 (1876), 93-98 (see p. 97) [From Tom Copeland, Sep 09 2010, vol number corrected Sep 10 2010]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, A commutative diagram of triangular arrays
E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005), 62-78.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, The Number of Convex Polyominoes with Given Height and Width, arXiv:1903.01095 [math.CO], 2019.
John H. Conway and N. J. A. Sloane, Low-dimensional lattices. VII Coordination sequences, Proc. R. Soc. Lond. A (1997) 453, 2369-2389.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005, 2008. [From _Peter Bala_, Oct 23 2008]
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 10, 13.
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9:2 (1997), pp. 319-335.
Weiping Wang and Tianming Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Yi Wang and Arthur L.B. Yang, Total positivity of Narayana matrices, arXiv:1702.07822 [math.CO], 2017.
Harold R. L. Yang and Philip B. Zhang, Stable multivariate Narayana polynomials and labeled plane trees, arXiv:2403.15058 [math.CO], 2024. See p. 2.

Row sums are in A000984. Columns 0-3 are A000012, A000290, A000537, A001249.
Family of polynomials (see A062145): this sequence (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Cf. A007318, A055133, A116647, A001263, A086645, A063007, A108558, A108625 (Hilbert transform), A145903, A181543, A086645 (logarithmic derivative), A105868 (inverse binomial transform), A093118.

Flat(List([0..10],n->List([0..n],k->Binomial(n,k)^2))); # Muniru A Asiru, Mar 30 2018

/* As triangle */ [[Binomial(n, k)^2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 15 2016

seq(seq(binomial(n, k)^2, k=0..n), n=0..10);

Table[Binomial[n, k]^2, {n, 0, 11}, {k, 0, n}]//Flatten (* Alonso del Arte, Dec 08 2013 *)

create_list(binomial(n,k)^2,n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

T(n,k):=if n=k then 1 else if k=0 then 1 else T(n-1,k)*(n+k)/(n-k)+T(n-1,k-1); /* Vladimir Kruchinin, Oct 18 2014 */

A(x,y):=1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2);
taylor(x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2),x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 23 2020 */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)^2)}; /* Michael Somos, May 03 2004 */

{T(n,k)=polcoeff(polcoeff(sum(m=0,n,(2*m)!/m!^2*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(2*m+1)),n,x),k,y)} \\ Paul D. Hanna, Oct 31 2010

def A008459(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2))**2 # Chai Wah Wu, Nov 12 2024

T(n,k) = A007318(n,k)^2. - Sean A. Irvine, Mar 29 2018
E.g.f.: exp((1+y)*x)*BesselI(0, 2*sqrt(y)*x). - Vladeta Jovovic, Nov 17 2003
G.f.: 1/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2); g.f. for row n: (1-t)^n P_n[(1+t)/(1-t)] where the P_n's are the Legendre polynomials. - Emeric Deutsch, Nov 23 2003 [The original version of the bivariate g.f. has been modified with the roles of x and y interchanged so that now x corresponds to n and y to k. - Petros Hadjicostas, Oct 22 2017]
G.f. for column k is Sum_{j = 0..k} C(k, j)^2*x^(k+j)/(1 - x)^(2*k+1). - Paul Barry, Nov 15 2005
Column k has g.f. (x^k)*Legendre_P(k, (1+x)/(1-x))/(1 - x)^(k+1) = (x^k)*Sum_{j = 0..k} C(k, j)^2*x^j/(1 - x)^(2*k+1). - Paul Barry, Nov 19 2005
Let E be the operator D*x*D, where D denotes the derivative operator d/dx. Then (1/n!^2) * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(2*n+1) = Sum_{k >= 0} binomial(n+k, k)^2*x^k. For example, when n = 3 we have (1/3!)^2*E^3(1/(1 - x)) = (1 + 9*x + 9*x^2 + x^3)/(1 - x)^7 = (1/3!)^2 * Sum_{k >= 0} ((k+1)*(k+2)*(k+3))^2*x^k. - Peter Bala, Oct 23 2008
G.f.: A(x, y) = Sum_{n >= 0} (2*n)!/n!^2 * x^(2*n)*y^n/(1 - x - x*y)^(2*n+1). - Paul D. Hanna, Oct 31 2010
From Peter Bala, Jul 24 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/n!^2 = BesselJ(0, 2*sqrt(-y)). Generating function: E(y)*E(x*y) = 1 + (1 + x)*y + (1 + 4*x + x^2)*y^2/2!^2 + (1 + 9*x + 9*x^2 + x^3)*y^3/3!^2 + .... Cf. the unsigned version of A021009 with generating function exp(y)*E(x*y).
The n-th power of this array has the generating function E(y)^n*E(x*y). In particular, the matrix inverse A055133 has the generating function E(x*y)/E(y). (End)
T(n,k) = T(n-1,k)*(n+k)/(n-k) + T(n-1,k-1), T(n,0) = T(n,n) = 1. - Vladimir Kruchinin, Oct 18 2014
Observe that the recurrence T(n,k) = T(n-1,k)*(n+k)/(n-k) - T(n-1,k-1), for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1 gives Pascal's triangle A007318. - Peter Bala, Dec 21 2014
n-th row polynomial R(n, x) = [z^n] (1 + (1 + x)*z + x*z^2)^n. Note that 1/n*[z^(n-1)] (1 + (1 + x)*z + x*z^2)^n gives the row polynomials of A001263. - Peter Bala, Jun 24 2015
Binomial transform of A105868. If G(x,t) = 1/sqrt(1 - 2*(1 + t)*x + (1 - t)^2*x^2) denotes the o.g.f. of this array then 1 + x*d/dx log(G(x,t)) = 1 + (1 + t)*x + (1 + 6*t + t^2)*x^2 + ... is the o.g.f. for A086645. - Peter Bala, Sep 06 2015
T(n,k) = Sum_{i=0..n} C(n-i,k)*C(n,i)*C(n+i,i)*(-1)^(n-i-k). - Vladimir Kruchinin, Jan 14 2018
G.f. satisfies A(x,y) = x*A(x,y)+x*y*A(x,y)+sqrt(1+4*x^2*y*A(x,y)^2). - Vladimir Kruchinin, Oct 23 2020
G.f. satisfies the differential equation y * d^2(A(x,y))/dy^2 - x^2 * d^2(x*A(x,y))/dx^2 + 2*x^2* A(x,y)^3 = 0. - Sergii Voloshyn, Mar 07 2025
T(n,k) = Sum_{i=0..n} C(2*n+1,i)*C(n+k-i,n)^2*(-1)^i. - Natalia L. Skirrow, Apr 14 2025

A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3x^k](1-x)^(3n+1), read by rows of k=0..2n terms.

Original entry on oeis.org

1, 1, 4, 1, 1, 20, 48, 20, 1, 1, 54, 405, 760, 405, 54, 1, 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1, 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1, 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1, 1, 490, 35623, 818300, 7917371, 37215794, 91789005, 123519792, 91789005, 37215794, 7917371, 818300, 35623, 490, 1, 1, 704, 73200, 2430400, 34657700, 246781248, 955910032, 2116980800, 2751843600, 2116980800, 955910032, 246781248, 34657700, 2430400, 73200, 704, 1

Offset: 0

Paul D. Hanna, Oct 30 2010

			Triangle begins:
 1;
 1, 4, 1;
 1, 20, 48, 20, 1;
 1, 54, 405, 760, 405, 54, 1;
 1, 112, 1828, 8464, 13840, 8464, 1828, 112, 1;
 1, 200, 5925, 52800, 182700, 273504, 182700, 52800, 5925, 200, 1;
 1, 324, 15606, 233300, 1424925, 4050864, 5703096, 4050864, 1424925, 233300, 15606, 324, 1; ...
Row g.f.s begin:
 (1) = (1-x)*(1 + x + x^2 + x^3 + x^4 +...);
 (1 + 4*x + x^2) = (1-x)^4*(1 + 2^3*x + 3^3*x^2 + 4^3*x^3 +...);
 (1 + 20*x + 48*x^2 + 20*x^3 + x^4) = (1-x)^7*(1 + 3^3*x + 6^3*x^2 +...);
 (1 + 54*x + 405*x^2 + 760*x^3 + 405*x^4 + 54*x^5 + x^6) = (1-x)^10*(1 + 4^3*x + 10^3*x^2 + 20^3*x^3 + 35^3*x^4 +...); ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, as a flattened triangle of rows 0..32
Ilia Gaiur, Vladimir Rubtsov, and Duco van Straten, Product formulas for the Higher Bessel functions, arXiv:2405.03015 [math.AG], 2024. See p. 18.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 14.

Cf. A181543, A181545, A006480, A002897, A000172.

Cf. A183204 (central terms), A183205.

t[n_, k_] := SeriesCoefficient[Sum[Binomial[n+j, j]^3*x^j, {j, 0, n+k}]*(1-x)^(3*n+1), {x,0, k}]; Table[t[n, k], {n, 0, 9}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Feb 04 2014, after PARI *)

{T(n,k)=polcoeff(sum(j=0,n+k,binomial(n+j,j)^3*x^j)*(1-x)^(3*n+1),k)}
for(n=0,10,for(k=0,2*n,print1(T(n,k),", "));print(""))

Row sums equal A006480(n) = (3n)!/(n!)^3, which is de Bruijn's s(3,n).
From Yahia Kahloune, Jan 30 2014: (Start)
Using these coefficients we can obtain formulas for the sums
Sum_{i=1..n} C(e-1+i,e)^3. Let us define b(k,e,3) = sum_{i=0..k-e} (-1)^i*C(3*e+1,i)*C(k-i,e)^3, where k=e+i.
For example:
b(e,e,3) = 1;
b(e+1,e,3) = (e+1)^3-(3*e+1) = e^2*(e+3);
b(e+2,e,3) = C(e+2,2)^3 - (3*e+1)*(e+1)^3 + C(3*e+1,2);
b(e+3,e,3) = C(e+3,e)^3 - (3*e+1)*C(e+2,e)^3 + C(3*e+1,2)*C(e+1,e)^3 - C(3*e+1,3);
b(e+4,e,3) = C(e+4,e)^3 - (3*e+1)*C(e+3,e)^3  + C(3*e+1,2)*C(e+2,e) - C(3*e+1,3)*C(e+1,e)^3 + C(3*e+1,4).
Then we have the formula: Sum_{i=1..n} C(e-1+i,e)^3 = Sum_{i=0..2*e} b(e+i,e,3)*C(n+e+i,3*e+1).
Example: Sum_{i=1..7} C(2+i,3)^3 = C(10,10) + 54*C(11,10) + 405*C(12,10) + 760*C(13,10) + 405*C(14,10) + 54*C(15,10) + C(16,10) = 820260. (End)
Let E be the operator D*x*D*x*D, where D denotes the derivative operator d/dx. Then (1/(n)!^3)  * E^n(1/(1 - x)) = (row n generating polynomial)/(1 - x)^(3*n+1) = Sum_{k >= 0} binomial(n+k, k)^3*x^k. For example, when n = 2 we have (1/2!)^3*E^3(1/(1 - x)) = (1 + 20 x + 48 x^2 + 20 x^3 + x^4)/(1 - x)^7. - Sergii Voloshyn, Dec 03 2024

A248658 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).

Original entry on oeis.org

1, 1, 1, 2, 9, 28, 66, 153, 433, 1345, 3952, 10991, 30954, 90988, 271845, 804153, 2361457, 6979690, 20842285, 62493914, 187274712, 561448399, 1688263179, 5093148285, 15393417178, 46570446829, 141063389488, 427979185898, 1300470246165, 3956367018001, 12048354848013, 36728336040306

Offset: 0

Paul D. Hanna, Oct 10 2014

nonn

Limit_{n->oo} a(n)/a(n+1) = 1 - t = t^3 = 0.3176721961... where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3).

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z) - (x*y*z)^3). - Ilya Gutkovskiy, Apr 23 2025

			G.f. A(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 66*x^6 + 153*x^7 +...
which equals the series:
A(x) = 1/(1-x-x^3) + 3!/1!^3*x^4/(1-x-x^3)^4 + 6!/2!^3*x^8/(1-x-x^3)^7 + 9!/3!^3*x^12/(1-x-x^3)^10 + 12!/4!^3*x^16/(1-x-x^3)^13 +...
The g.f. also equals the series:
A(x) = 1 +
x*(1 + x^2) +
x^2*(1 + 2^3*x^2 + x^4) +
x^3*(1 + 3^3*x^2 + 3^3*x^4 + x^6) +
x^4*(1 + 4^3*x^2 + 6^3*x^4 + 4^3*x^6 + x^8) +
x^5*(1 + 5^3*x^2 + 10^3*x^4 + 10^3*x^6 + 5^3*x^8 + x^10) +...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A181545, A181543.

Table[Sum[Binomial[n-2*k,k]^3,{k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2014 *)

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^(2*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n,x^(4*m)/(1-x-x^3 +x*O(x^n))^(3*m+1)*(3*m)!/(m!)^3), n)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=sum(k=0, n\3, binomial(n-2*k, k)^3)}
for(n=0, 40, print1(a(n), ", "))

G.f.: A(x) = Sum_{n>=0} (3*n)!/(n!)^3 * x^(4*n) / (1-x-x^3)^(3*n+1).
a(n) = Sum_{k=0..[n/3]} C(n-2*k,k)^3.
G.f.: A(x) = G( x^4/(1-x-x^3)^3 )/(1-x-x^3) where G(x) satisfies:
* G(x^3) = G( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x) and G(x) is the g.f. of A006480.

A183065 Triangle defined by g.f.: Sum_{n>=0} (4n)!/n!^4 x^(2n)y^n/(1-x-xy)^(4n+1), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 26, 1, 1, 123, 123, 1, 1, 364, 3246, 364, 1, 1, 845, 25210, 25210, 845, 1, 1, 1686, 120135, 606500, 120135, 1686, 1, 1, 3031, 430941, 6082475, 6082475, 430941, 3031, 1, 1, 5048, 1277668, 38698856, 137915470, 38698856, 1277668, 5048, 1, 1, 7929

Offset: 0

Paul D. Hanna, Dec 22 2010

Compare the g.f. of this triangle with the g.f.s of triangles:
* A008459: Sum_{n>=0} (2n)!/n!^2 * x^(2n)*y^n/(1-x-xy)^(2n+1),
* A181543: Sum_{n>=0} (3n)!/n!^3 * x^(2n)*y^n/(1-x-xy)^(3n+1),
which have terms A008459(n,k) = C(n,k)^2 and A181543(n,k) = C(n,k)^3.

			G.f.: A(x,y) = 1/(1-x-xy) + 4!*x^2*y/(1-x-xy)^5 + (8!/2!^4)*x^4*y^2/(1-x-xy)^9 + (12!/3!^4)*x^6*y^3/(1-x-xy)^13 +...
Triangle begins:
1;
1, 1;
1, 26, 1;
1, 123, 123, 1;
1, 364, 3246, 364, 1;
1, 845, 25210, 25210, 845, 1;
1, 1686, 120135, 606500, 120135, 1686, 1;
1, 3031, 430941, 6082475, 6082475, 430941, 3031, 1;
1, 5048, 1277668, 38698856, 137915470, 38698856, 1277668, 5048, 1; ...

Cf. A183066 (column 1), A183067 (row sums), A183068 (central terms).

{T(n,k)=polcoeff(polcoeff(sum(m=0,n,(4*m)!/m!^4*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(4*m+1)),n,x),k,y)}

A193420 -log( Sum_{n>=0} (-x)^n/n!^3 ) = Sum_{n>=1} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 3, 46, 1899, 163476, 25333590, 6412369860, 2473269931755, 1379817056827720, 1069150908119474628, 1113779885682143602440, 1518901247410616194635510, 2651993653876241574715172280, 5817640695573490720735010689620

Offset: 1

Paul D. Hanna, Jul 26 2011

nonn

			L(x) = -log(1 - x + x^2/2!^3 - x^3/3!^3 + x^4/4!^3 - x^5/5!^3 +-...)
where
L(x) = x + 3*x^2/2!^3 + 46*x^3/3!^3 + 1899*x^4/4!^3 + 163476*x^5/5!^3 +...
ALTERNATE GENERATING METHOD.
A signed version of A181543(n,k) = C(n,k)^3 begins:
1;
1, 1;
1, 8, 1;
1, 27, 27, 1;
1, 64, 216, 64, 1;
1, 125, 1000, 1000, 125, 1; ...
The matrix log of triangle A181543 begins:
0;
1, 0;
-3, 8, 0;
46, -81, 27, 0;
-1899, 2944, -648, 64, 0;
163476, -237375, 46000, -3000, 125, 0; ...
in which this sequence (signed) is found in column 0.

Cf. A002190 (variant), A181543.

{a(n)=n!^3*polcoeff(-log(sum(m=0,n,(-x)^m/m!^3)+x*O(x^n)),n)}

/* As Column 0 of the Matrix Log of signed Triangle A181543 */
{a(n)=local(L,M=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*binomial(r-1,c-1)^3)));
L=sum(n=1,#M,(M^0-M)^n/n);if(n<0,0,L[n+1,1])}

Equals column 0 of the matrix log of triangle T(n,k) = (-1)^(n-k)*C(n,k)^3.

a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^3 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021

A186097 G.f. satisfies: A(x) = Sum_{n>=0} x^n[Sum_{k=0..n} C(n,k)^3 x^k* A(x)^k].

Original entry on oeis.org

1, 1, 2, 10, 39, 147, 639, 2857, 12725, 58081, 270250, 1268444, 6009439, 28736727, 138401100, 670641714, 3268021317, 16004012529, 78716657052, 388701645264, 1926266491659, 9576792342099, 47753368809171, 238759903786041

Offset: 0

Paul D. Hanna, Feb 12 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 39*x^4 + 147*x^5 + 639*x^6 +...
where g.f. A(x) satisfies:
* A(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + 8*x*A(x) + x^2*A(x)^2) + x^3*(1 + 27*x*A(x) + 27*x^2*A(x)^2 + x^3*A(x)^3) + x^4*(1 + 64*x*A(x) + 216*x^2*A(x)^2 + 64*x^3*A(x)^3 + x^4*A(x)^4) +...;

Cf. A181665, A183876, A181543.

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^3*x^k))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3,(3*m)!/m!^3*x^(3*m)*A^m/(1-x-x^2*A+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

G.f. satisfies:
(1) A(x) = Sum_{n>=0} x^(2n)*A(x)^n*[Sum_{k>=0} C(n+k,k)^3*x^k].
(2) A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(3n)*A(x)^n/(1-x-x^2*A(x))^(3n+1).

A320824 T(n, k) = (mn)!/(k!(n-k)!)^m with m = 3; triangle read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 6, 6, 90, 720, 90, 1680, 45360, 45360, 1680, 34650, 2217600, 7484400, 2217600, 34650, 756756, 94594500, 756756000, 756756000, 94594500, 756756, 17153136, 3705077376, 57891834000, 137225088000, 57891834000, 3705077376, 17153136

Offset: 0

Peter Luschny, Oct 21 2018

			Triangle starts:
[0]      1;
[1]      6,        6;
[2]     90,      720,        90;
[3]   1680,    45360,     45360,      1680;
[4]  34650,  2217600,   7484400,   2217600,    34650;
[5] 756756, 94594500, 756756000, 756756000, 94594500, 756756;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A007318 (Pascal, m=1), A069466 (m=2), this sequence (m=3).

Cf. A006480, A181543.

Flat(List([0..6],n->List([0..n],k->Factorial(3*n)/(Factorial(k)*Factorial(n-k))^3))); # Muniru A Asiru, Oct 27 2018

[[(Factorial(3*n)/(Factorial(n))^3)*Binomial(n, k)^3: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Oct 27 2018

T := (n, k, m) -> (m*n)!/(k!*(n-k)!)^m:
seq(seq(T(n, k, 3), k=0..n), n=0..7);

Table[((3*n)!/(n!)^3)*Binomial[n, k]^3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2018 *)

t(n, k) = (3*n)!/(k!*(n-k)!)^3
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 6 rows of triangle as follows: */
trianglerows(6) \\ Felix Fröhlich, Oct 21 2018

T(n, k) = ((3*n)!/(n!)^3) * binomial(n, k)^3 = A006480(n)*A181543(n, k).

A200212 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^kA(x)^(n-k)] x^n/n ).

Original entry on oeis.org

1, 1, 3, 11, 42, 174, 763, 3457, 16075, 76351, 368767, 1805682, 8943948, 44736096, 225646033, 1146461185, 5862224756, 30144922281, 155791900727, 808773877919, 4215675455503, 22054576750972, 115765182718467, 609508331610920, 3218059655553030, 17034314889643633

Offset: 0

Paul D. Hanna, Nov 14 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 42*x^4 + 174*x^5 + 763*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^3*x*A + x^2)*x^2/2 +
(A^3 + 3^3*x*A^2 + 3^3*x^2*A + x^3)*x^3/3 +
(A^4 + 4^3*x*A^3 + 6^3*x^2*A^2 + 4^3*x^3*A + x^4)*x^4/4 +
(A^5 + 5^3*x*A^4 + 10^3*x^2*A^3 + 10^3*x^3*A^2 + 5^3*x^4*A + x^5)*x^5/5 +
(A^6 + 6^3*x*A^5 + 15^3*x^2*A^4 + 20^3*x^3*A^3 + 15^3*x^4*A^2 + 6^3*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 117*x^4/4 + 581*x^5/5 + 2987*x^6/6 + 15499*x^7/7 + 81213*x^8/8 +...

Cf. A192131, A166896, A198944, A199875, A166990, A198950, A181143, A181543, A200074.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

cp's OEIS Frontend

A216207 Matrix inverse of A181543 (cubes of entries of Pascal's triangle).

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

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A008459 Square the entries of Pascal's triangle.

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A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3*x^k]*(1-x)^(3n+1), read by rows of k=0..2n terms.

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A248658 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^3 * x^(2*k).

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A183065 Triangle defined by g.f.: Sum_{n>=0} (4*n)!/n!^4 * x^(2*n)*y^n/(1-x-x*y)^(4*n+1), read by rows.

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A181544 Triangle in which the g.f. for row n is [Sum_{k>=0} C(n+k-1,k)^3x^k](1-x)^(3n+1), read by rows of k=0..2n terms.

A183065 Triangle defined by g.f.: Sum_{n>=0} (4n)!/n!^4 x^(2n)y^n/(1-x-xy)^(4n+1), read by rows.

A186097 G.f. satisfies: A(x) = Sum_{n>=0} x^n[Sum_{k=0..n} C(n,k)^3 x^k* A(x)^k].

A320824 T(n, k) = (mn)!/(k!(n-k)!)^m with m = 3; triangle read by rows, 0 <= k <= n.

A200212 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^kA(x)^(n-k)] x^n/n ).