A268553 -id:A268553 - cp's OEIS frontend

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

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

A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

Original entry on oeis.org

30, 6218, 2658432, 1054788750, 552306591900, 269380692717696, 155175092086118400, 83798883891736779150, 50885239237727996887500, 29198209396114625497699068, 18332853214682572877389897728, 10951674446687597386319569942656, 7036938452279110885561897815723264, 4325988198220149508865311059521280000

Offset: 1

Janaka Rodrigo, Nov 18 2022

nonn

Alternatively a(n) is the number of ways two ants can interchange their positions starting simultaneously from two opposite corners and moving along shortest grid paths at same speed in an n X n X n grid without meeting other one.

			When n=2 number of ways to move between two opposite corners are given by 6!/(2!*2!*2!) and number of such pairs are given by (6!/(2!*2!*2!))^2. This total number of pairs are given by A268553(2)=8100.
Number of pairs which have the middle point of their paths as a common point are A357760(2)=1782.
Therefore number of pairs without having middle point on their paths as a common point are 8100-1782=6218

Cf. A268553, A357760.

a(n) = A268553(n) - A357760(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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