A001448 -id:A001448 - cp's OEIS frontend

A387248 a(n) = 3/(n + 1) * Catalan(2*n).

3, 3, 14, 99, 858, 8398, 89148, 1002915, 11785890, 143291610, 1790214660, 22870640910, 297670187844, 3935861372604, 52749590350072, 715309969142307, 9800129095949682, 135490673691621794, 1888389218820071604, 26510079418051005210, 374589577468070301260, 5324240442532424176260, 76082624294738699098440

Offset: 0

Peter Bala, Aug 24 2025

Bisection of A007054.
Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).
For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Paolo Xausa, Table of n, a(n) for n = 0..800

Cf. A000108, A000984, A001448, A007054, A387249, A387250.

seq( 6/((2*n+1)*(2*n+2)) * binomial(4*n, 2*n), n = 0..22);

A387248[n_] := 3*CatalanNumber[2*n]/(n + 1); Array[A387248, 25, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = 6/((2*n + 1)*(2*n + 2)) * binomial(4*n, 2*n).
a(n) = 4*Catalan(2*n) - Catalan(2*n+1) (showing a(n) to be an integer)
G.f.: A(x) = ((2 - f(x))*sqrt(2 + 2*f(x)) - 2)/(4*x), where f(x) = sqrt(1 - 16*x).
a(n) = 2*(4*n - 1)*(4*n - 3)/((n + 1)*(2*n + 1)) * a(n-1) with a(0) = 3.
a(n) ~ 3/(2*sqrt(2*Pi)) * 16^n/n^(5/2).
a(n) is odd iff n = 2^k - 1 for some k, so a(n) has the same parity as Catalan(n).
E.g.f.: 3*hypergeom([1/4, 3/4], [3/2, 2], 16*x). - Stefano Spezia, Aug 27 2025

A073077 Numbers k that divide C(2k,k) and C(4k,2k).

Original entry on oeis.org

1, 2, 20, 110, 156, 210, 220, 238, 240, 312, 460, 468, 483, 510, 561, 600, 624, 665, 684, 696, 720, 744, 770, 806, 812, 816, 868, 936, 966, 1001, 1012, 1045, 1064, 1100, 1110, 1122, 1144, 1155, 1170, 1200, 1295, 1309, 1320, 1326, 1332, 1360, 1368, 1394, 1404

Offset: 1

Benoit Cloitre, Aug 17 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A014847.

Cf. A000984, A001448.

Select[Range[1500], Divisible[Binomial[2*#, #], #] && Divisible[Binomial[4*#, 2*#], #] &] (* Amiram Eldar, Apr 26 2025 *)

isok(n) = !(binomial(2*n, n) % n) && !(binomial(4*n, 2*n) % n); \\ Michel Marcus, Nov 28 2013

More terms from Michel Marcus, Nov 28 2013

A122882 Array of T(n,m)=15...(4n-3)37...(4m-1)2^(n+m)/(n+m)! by antidiagonals.

Original entry on oeis.org

1, 2, 6, 10, 6, 42, 60, 20, 28, 308, 390, 90, 70, 154, 2310, 2652, 468, 252, 308, 924, 17556, 18564, 2652, 1092, 924, 1540, 5852, 134596, 132600, 15912, 5304, 3432, 3960, 8360, 38456, 1038312, 961350, 99450, 27846, 14586, 12870, 18810, 48070

Offset: 0

Michael Somos, Sep 16 2006

T(n,m)=2*A(m,n) in Problem A10527 Solution.

			       1        6       42      308     2310    17556 ...
       2        6       28      154      924     5852 ...
      10       20       70      308     1540     8360 ...
      60       90      252      924     3960    18810 ...
     390      468     1092     3432    12870    54340 ...
    2652     2652     5304    14586    48620   184756 ...
   18564    15912    27846    68068   204204   705432 ...
  132600    99450   154700   340340   928200  2939300 ...
  961350   640900   897260  1794520  4486300 13113800 ...
 7049900  4229940  5383560  9869860 22776600 61822200 ...

V. Pasol, Problem 10527, Amer. Math. Monthly, 103 (1996), p. 427; Is it an integer: solution to problem 10527, Amer. Math. Monthly, 104 (1997), 980-981.

Cf. A004981(n)=T(n, 0), A004982(n)=T(0, n), A001448(n)=T(n, n).

A122882 := proc(n,m)
    mul(4*i-3,i=1..n)*mul(4*i-1,i=1..m) ;
    %*2^(n+m)/(n+m)! ;
end proc: # R. J. Mathar, Sep 24 2021

{T(n,m)=if(n<0||m<0, 0, 2^(n+m)/(n+m)!*prod(k=1, m, 4*k-1)*prod(k=1, n, 4*k-3))}

T(n,m) = T(n,m-1)*(8*m-2)/(n+m) = T(n-1,m)*(8*n-6)/(n+m). T(0,0) = 1.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.

Original entry on oeis.org

1, 48, 15120, 7392000, 4414410000, 2956651746048, 2133278987583744, 1621682968820428800, 1281351259836532170000, 1043032815185819858400000, 869343653096068540955685120, 738637974389826550020188712960, 637665137404661719206664998969600

Offset: 0

Bradley Klee, Jun 04 2019

nonn

Entry number six in the "Big Table" of Almkvist et al. (see links). The period T(x) = Sum_{n>=0} a(n)*x^(2*n) is also the first x-derivative of the 6-volume associated to the algebraic variety V6 = P1 & P2 & P3, with P1 : X1^2 + Y1^2 = X2^2 + Y2^2, P2 : X2^2 + Y2^2 = X3^2 + Y3^2, P3 : x=(X1^2 + X2^2 + X3^2 + Y1^2 + Y2^2 + Y3^2)^3*(1 - X1*X2*X3*Y1*Y2*Y3). The small x limit reduces V6 to a 6-ball with 6-volume proportional to x. Similar constructions are known to exist for a few other geometries on Almkvist's list, most notably #3: A186420, and #16: A039699.

G. Almkvist et al., Tables of Calabi-Yau Equations, arXiv:math/0507430 [math.AG], 2005-2010, p. 10.
G. Almkvist et al., Raw Calabi-Yau Period Data, Johannes Gutenberg Universität Mainz, 2019, case 1.6.

Hadamard Factors: A000984, A002894, A002897, A001448, A000897, A008977.

Calabi-Yau Periods: A008978, A186420, A268553, A039699.

Binomial[4*#,2*#]*Binomial[2*#,#]^3&/@Range[0,10]

G.f.: 4F3({1/4, 3/4, 1/2, 1/2}, {1, 1, 1}, 1024*x).
Define the period integral:
dt(x) = dz1*dz2*dz3/sqrt(1-32*x*cos(z1)*cos(z2)*cos(z3)).
T(x)=1/(2*Pi)^3*Integral_{0..2*Pi,0..2*Pi,0..2*Pi} dt(x),
the Picard-Fuchs coefficients:(c0,c1,c2,c3,c4)=
(768*x, 14592*x^2-1, x*(25344*x^2-7), 2*x^2*(5120*x^2-3), x^3*(32*x-1)*(32*x+1)),
and the certificate function:
G(z1,z2,z3)=(16*sin(z1)*(
       48*x*cos(z1)
     + cos(z2)*cos(z3)
     + 48*x*cos(z1)*(cos(z3)^2 + cos(z2)^2)
     + 2304*x^2*cos(z1)^2*cos(z2)*cos(z3)
     + 80*x*cos(z1)*cos(z2)^2*cos(z3)^2
     + 384*x^2*cos(z1)^2*(cos(z2)*cos(z3)^3 + cos(z2)^3*cos(z3))
     + 256*x^2*cos(z1)^2*cos(z2)^3*cos(z3)^3)
    )/(3*(1 - 32*x*cos(z1)*cos(z2)*cos(z3))^(7/2)),
Then: 0 = Sum_{n=0..4}cn*d^n/dx^n dt(x) + d/dz1 G(z1,z2,z3) + d/dz2 G(z2,z3,z1) + d/dz3 G(z3,z1,z2), thus: 0 = Sum_{n=0..4} cn*d^n/dx^n T(x).
Furthermore, let (a1,a2,a3)=(c1,c2,c3)/c0, then also: 0 = (1/2)*a2*a3 - (1/8)*a3^3 + d/dx(a2) - (3/4)*a3*d/dx(a3) - (1/2)*d^2/dx^2(a3) - a1.
D-finite with recurrence: n^4*a(n) -16*(4*n-1)*(4*n-3)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jan 27 2020

A359761 a(n) = binomial(4n, 2n)(2n)!/(2^n*n!).

Original entry on oeis.org

1, 6, 210, 13860, 1351350, 174594420, 28109701620, 5421156741000, 1218404977539750, 312723944235202500, 90252130306279441500, 28929910132721937339000, 10197793321784482911997500, 3920659309406065045704885000, 1632674555274097086889962825000, 732091270584905133761459330730000

Offset: 0

Peter Luschny, Jan 14 2023

Cf. A001147, A001448, A359760.

a := binomial(4*n, 2*n)*(2*n)!/(2^n*n!):
seq(a(n), n = 0..15);

a[n_] := Binomial[4*n, 2*n]*(2*n)!/(2^n*n!); Array[a, 20, 0] (* Amiram Eldar, Sep 05 2025 *)

a(n) = (2^(3*n)*Gamma(2*n + 1/2))/(sqrt(Pi)*Gamma(n + 1)).
a(n) = A359760(4*n, 2*n), the central terms of the triangle without the zeros.
From R. J. Mathar, Jan 25 2023: (Start)
a(n) = A001448(n)*A001147(n).
D-finite with recurrence n*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. (End)
From Stefano Spezia, Aug 24 2025: (Start)
E.g.f.: 2*EllipticK(8*sqrt(2*x)/(1 + 4*sqrt(2*x)))/(Pi*sqrt(1 + 4*sqrt(2*x))).
E.g.f.: hypergeom([1/2, 1/2], [1], 8*sqrt(2*x)/(1 + 4*sqrt(2*x)))/sqrt(1 + 4*sqrt(2*x)). (End)
a(n) ~ (32/e)^n * n^(n-1/2) / sqrt(Pi). - Amiram Eldar, Sep 05 2025

A349468 a(n) = (4n)! / (n! (2*n)!).

Original entry on oeis.org

1, 12, 840, 110880, 21621600, 5587021440, 1799020903680, 693908062848000, 311911674250176000, 160114659448423680000, 92418181433630148096000, 59248455951814527670272000, 41770161446029242007541760000, 32118041062654484854414417920000, 26749739913610806671605150924800000

Offset: 0

Ilya Gutkovskiy, Nov 18 2021

Cf. A000897, A000984, A001448, A001813, A009120, A166338.

Table[(4 n)!/(n! (2 n)!), {n, 0, 14}]
nmax = 14; CoefficientList[Series[2 EllipticK[16 Sqrt[x]/(1 + 8 Sqrt[x])]/(Pi Sqrt[1 + 8 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[SeriesCoefficient[D[1/Sqrt[1 - 4 x], {x, n}], {x, 0, n}], {n, 0, 14}]

a(n) = (4*n)! / (n! * (2*n)!) \\ Andrew Howroyd, Nov 20 2021

E.g.f.: 2 * EllipticK( 16*sqrt(x) / (1 + 8*sqrt(x)) ) / (Pi * sqrt(1 + 8*sqrt(x))).
a(n) is the coefficient of x^n in expansion of d^n/dx^n g(x), where g(x) is the g.f. of central binomial coefficients (A000984).
a(n) = n! * A000897(n) = A009120(n) / n! = A166338(n) / (2*n)! = A001448(n) * A001813(n).
a(n) ~ 64^n * n^(n-1/2) / (sqrt(Pi) * exp(n)).
D-finite with recurrence n*a(n) -4*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A387248 a(n) = 3/(n + 1) * Catalan(2*n).

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A073077 Numbers k that divide C(2k,k) and C(4k,2k).

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A122882 Array of T(n,m)=1*5*...*(4n-3)*3*7*...*(4m-1)*2^(n+m)/(n+m)! by antidiagonals.

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A307618 A Calabi-Yau period integral: a(n) = C(4*n,2*n)*C(2*n,n)^3.

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A359761 a(n) = binomial(4*n, 2*n)*(2*n)!/(2^n*n!).

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Maple

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Formula

A349468 a(n) = (4*n)! / (n! * (2*n)!).

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Mathematica

PARI

Formula

A122882 Array of T(n,m)=15...(4n-3)37...(4m-1)2^(n+m)/(n+m)! by antidiagonals.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.

A359761 a(n) = binomial(4n, 2n)(2n)!/(2^n*n!).

A349468 a(n) = (4n)! / (n! (2*n)!).