A188662 -id:A188662 - cp's OEIS frontend

A275047 Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only].

1, 18, 1350, 141120, 17151750, 2272538268, 318430816704, 46404203788800, 6961609406993670, 1068002895589987500, 166779781860762170100, 26422986893371642828800, 4236593267629481817240000, 686167053247777413372681600, 112093956900827388909570240000

Offset: 0

Gheorghe Coserea, Jul 18 2016

Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,-z,w), where R(x,y,z,w) = 1/(1-(1+w)*(x*y + x*z + y*z)).
From Peter Bala, Jun 22 2023: (Start)
a(n) = A(n,n,2*n,2*n) (= A(2*n,2*n,n,n)) in the notation of Straub, equation 8, where it is shown that the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r. This also follows from Meštrović equation 39, since a(n) = binomial(3*n,n)^2 * binomial(2*n,n).
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n.
We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5, and positive integers n and r. Cf. A362725 and A362732. (End)

			1 + 18*x^2 + 1350*x^4 + 141120*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..444 (first 34 terms from Gheorghe Coserea)
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.
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], 2011.
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A000984, A188662, A268545 - A268555.

a:= proc(n) option remember; `if`(n=0, 1,
      9*(3*n-1)^2*(3*n-2)^2*a(n-1)/((4*n-2)*n^3))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

Table[(3*n)!^2 / (n!^4*(2*n)!), {n, 0, 20}] (* Vaclav Kotesovec, Aug 03 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1/2, 1, 1}, 729x/4], {x, 0, 10}], x] (* Benedict W. J. Irwin, Aug 05 2016 *)

my(x='x, y='y, z='z, w='w);
R = 1/(1-(1+w)*(x*y+x*z+y*z));
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(23, R, [x,y,z,w])

0 = (-4*x^2+729*x^4)*y'''' + (-20*x+7290*x^3)*y''' + (-16+18063*x^2)*y'' + 10449*x*y' + 576*y, where y = 1 + 18*x^2 + 1350*x^4 + ...
From Vaclav Kotesovec, Aug 03 2016: (Start)
a(n) = (3*n)!^2 / (n!^4 * (2*n)!).
a(n) ~ 3^(6*n+1) / (Pi^(3/2) * n^(3/2) * 2^(2*n+2)).
(End)
G.f.: 4F3(1/3,1/3,2/3,2/3;1/2,1,1;729x/4). - Benedict W. J. Irwin, Aug 05 2016
From Peter Bala, Sep 20 2021: (Start)
a(n) = 9*(3*n - 1)^2*(3*n - 2)^2/(2*n^3*(2*n - 1))*a(n-1).
a(n) = Sum_{k = n..3*n} (-1)^k*binomial(3*n,k)^2*binomial(k,n)^2. (End)
From Peter Bala, Jun 22 2023: (Start)
a(n) = binomial(3*n,n)^2 * binomial(2*n,n) = A188662(n) * A000984(n).
a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(2*n,k)*binomial(2*n-k,n)* binomial(4*n-k,2*n).
a(n) = [(x*y)^n * (z*t)^(2*n)] 1/((1 - x - y)*(1 - z - t) - x*y*z*t). (End)

A361885 a(n) = (1/n) * Sum_{k = 0..2n} (n+2k) * binomial(n+k-1,k)^3.

Original entry on oeis.org

9, 979, 165816, 33372819, 7380882509, 1732912534168, 424032181044264, 106952563532680339, 27609695174536836075, 7259294757681340436979, 1937215339689731617386000, 523352118643145676922317336, 142854011885066484369862826496, 39337931825265398967484384872560

Offset: 0

Peter Bala, Mar 28 2023

Compare with the closed form evaluation of the binomial sums (1/n) * Sum_{k = 0..2*n} (-1)^k * (n + 2*k) * binomial(n+k-1,k) = binomial(3*n,n) and (1/n) * Sum_{k = 0..2*n} (n + 2*k) * binomial(n+k-1,k)^2 = binomial(3*n,n)^2.
The binomial coefficients u(n) := binomial(3*n,n) = A005809(n) satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for positive integers n and r and all primes p >= 5. We conjecture that the present sequence satisfies the same congruences.
More generally, for m >= 3, the sequences {b_m(n) : n >= 1} and {c_m(n) : n >= 1} defined by b_m(n) = (1/n) * Sum_{k = 0..2*n} (n + 2*k) * binomial(n+k-1,k)^m and c_m(n) = (1/n) * Sum_{k = 0..2*n} (-1)^k * (n + 2*k) * binomial(n+k-1,k)^m may satisfy the same congruences.

Cf. A005809, A188662, A361883, A361884, A361886.

seq( (1/n)*add((n + 2*k) * binomial(n+k-1,k)^3, k = 0..2*n), n = 1..20);

Table[Sum[(n+2*k) * Binomial[n+k-1,k]^3, {k,0,2*n}]/n, {n,1,20}] (* Vaclav Kotesovec, Mar 29 2023 *)

a(n) = (1/n) * sum(k = 0, 2*n,  (n+2*k) * binomial(n+k-1,k)^3); \\ Michel Marcus, Mar 30 2023

a(n) ~ 5 * 3^(9*n + 3/2) / (19 * Pi^(3/2) * n^(3/2) * 2^(6*n + 3)). - Vaclav Kotesovec, Mar 29 2023

A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

Original entry on oeis.org

1, 2, 2, 6, 9, 6, 20, 40, 40, 20, 70, 175, 225, 175, 70, 252, 756, 1176, 1176, 756, 252, 924, 3234, 5880, 7056, 5880, 3234, 924, 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432, 12870, 57915, 135135, 212355, 245025, 212355, 135135, 57915, 12870

Offset: 0

Peter Luschny, Mar 21 2024

The main diagonal and column 0 of the triangle are the central binomial coefficients, which are the sums of the squares of Pascal's triangle entries. This sum representation can be generalized, and all terms can be seen as sums of coefficients of some polynomials. (See the Example section.)

To see this, consider T(n, k) as the value of the polynomials P(n, k)(x) at x = 1, where P(n, k)(x) = H([-n, -k], [1], x)*H([-n, -n + k], [1], x) and H denotes the hypergeometric sum 2F1. For instance column 0 is given by the row sums of A008459, and column 1 by the row sums of A371401.

			Triangle starts:
[0]    1;
[1]    2,     2;
[2]    6,     9,     6;
[3]   20,    40,    40,    20;
[4]   70,   175,   225,   175,    70;
[5]  252,   756,  1176,  1176,   756,   252;
[6]  924,  3234,  5880,  7056,  5880,  3234,   924;
[7] 3432, 13728, 28512, 39600, 39600, 28512, 13728, 3432;
.
Because of the symmetry, only the sum representation of terms with k <= n/2 are shown.
0:                 [1]
1:               [1+1]
2:             [1+4+1],               [1+4+4]
3:           [1+9+9+1],            [1+9+21+9]
4:      [1+16+36+16+1],       [1+16+66+76+16],        [1+16+76+96+36]
5: [1+25+100+100+25+1], [1+25+160+340+205+25], [1+25+190+460+400+100]

Column 0 and main diagonal are A000984.
Column 1 and subdiagonal are A097070.
Row sums are A045721.
The even bisection of the alternating row sums is A005809.
The central terms are A188662.
Cf. A046899, A092392, A371395, A244038, A371399.
Cf. A008459, A371401.

T := (n, k) -> binomial(k + n, k) * binomial(2*n - k, n):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

T[n_, k_] := Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, -n +k, 1, 1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]

T(n, k) = A046899(n, k) * A092392(n, k).
T(n, k) = A046899(n, k) * A046899(n, n - k).
T(n, k) = A092392(n, k) * A092392(n, n - k).
T(n, k) = A371395(n, k) * (n + 1).
T(n, k) = hypergeom([-n, -k], [1], 1) * hypergeom([-n, -n + k], [1], 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A244038(n).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A371399(n).

A277584 a(n) = binomial(3n-1, n-1)^2.

Original entry on oeis.org

0, 1, 25, 784, 27225, 1002001, 38291344, 1502337600, 60101954649, 2440703175625, 100300325150025, 4161829109817600, 174077451630810000, 7330421677037621904, 310467090932230849600, 13214837914326197526784, 564927069263895118093401

Offset: 0

Seiichi Manyama, Oct 22 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A025174, A060150, A188662.

[Binomial(3*n-1, n-1)^2: n in [0..20]]; // Vincenzo Librandi, Oct 23 2016

Table[Boole[n > 0] Binomial[3 n - 1, n - 1]^2, {n, 0, 16}] (* Michael De Vlieger, Oct 26 2016 *)

a(n) = binomial(3*n-1, n-1)^2; \\ Michel Marcus, Oct 22 2016

a(n) = A025174(n)^2.
a(n) = A188662(n)/9 for n > 0.
Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
a(n) = (Sum_{k=0..n} binomial(n, k)^2 * ((2*n, 2*n - k)))/5 for n > 0.

A370232 Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 36, 25, 1, 1, 100, 225, 49, 1, 1, 225, 1225, 784, 81, 1, 1, 441, 4900, 7056, 2025, 121, 1, 1, 784, 15876, 44100, 27225, 4356, 169, 1, 1, 1296, 44100, 213444, 245025, 81796, 8281, 225, 1, 1, 2025, 108900, 853776, 1656369, 1002001, 207025, 14400, 289, 1

Offset: 0

Peter Luschny, Feb 12 2024

			Triangle starts:
[0] 1;
[1] 1,     1;
[2] 1,     9,       1;
[3] 1,    36,      25,       1;
[4] 1,   100,     225,      49,       1;
[5] 1,   225,    1225,     784,      81,      1;
[6] 1,   441,    4900,    7056,    2025,    121,     1;
[7] 1,   784,   15876,   44100,   27225,   4356,   169,   1;

Shifted bisection of A182878.

Cf. A370233 (c=2), A188648 (row sums), A188662 (central terms).

Table[Binomial[n + k, 2*k]^2, {n, 0, 7}, {k, 0, n}] // Flatten

T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 1.

T(n, k) = [z^k] hypergeom([-n, -n, 1 + n, 1 + n], [1/2, 1/2, 1], z/16).

A322189 G.f. A(x) satisfies: A(x)^2 + A(x) - 1 = Sum_{n>=0} binomial(3n,n)^2 x^n.

Original entry on oeis.org

1, 3, 72, 2208, 75531, 2748957, 104125542, 4055630148, 161248468944, 6513248563281, 266402605165194, 11007646816287168, 458676184166135532, 19248392999470239126, 812657808793768897362, 34489498873811554580556, 1470421670132406007539195, 62941195430565633995463225, 2703764557673857477236184014, 116513978125127785773539029596

Offset: 0

Paul D. Hanna, Dec 07 2018

nonn

Radius of convergence of g.f. A(x) is r = 2^4/3^6 = 16/729.

			G.f.: A(x) = 1 + 3*x + 72*x^2 + 2208*x^3 + 75531*x^4 + 2748957*x^5 + 104125542*x^6 + 4055630148*x^7 + 161248468944*x^8 + 6513248563281*x^9 + ...
such that
A(x)^2 + A(x) - 1 = 1 + 9*x + 225*x^2 + 7056*x^3 + 245025*x^4 + 9018009*x^5 + 344622096*x^6 + 13521038400*x^7 + 540917591841*x^8 + 21966328580625*x^9 + ... + binomial(3*n,n)^2 * x^n + ...

Cf. A188662.

{S(n) = sum(m=0,n, binomial(3*m,m)^2 * x^m ) +x*O(x^n)}
{A(n) = (sqrt(4*S(n) + 5) - 1)/2 }
{a(n) = polcoeff( A(n), n)}
for(n=0,30, print1(a(n),", "))

cp's OEIS Frontend

A275047 Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only].

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A361885 a(n) = (1/n) * Sum_{k = 0..2*n} (n+2*k) * binomial(n+k-1,k)^3.

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A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)*binomial(2*n - k, n).

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A277584 a(n) = binomial(3n-1, n-1)^2.

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A370232 Triangle read by rows. T(n, k) = binomial(n + k, 2*k)^2.

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A322189 G.f. A(x) satisfies: A(x)^2 + A(x) - 1 = Sum_{n>=0} binomial(3*n,n)^2 * x^n.

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A361885 a(n) = (1/n) * Sum_{k = 0..2n} (n+2k) * binomial(n+k-1,k)^3.

A371400 Triangle read by rows: T(n, k) = binomial(k + n, k)binomial(2n - k, n).

A322189 G.f. A(x) satisfies: A(x)^2 + A(x) - 1 = Sum_{n>=0} binomial(3n,n)^2 x^n.