cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A187056 G.f.: A(x,y,z) = Sum_{n>=0} ((2n)!/n!^2)*[Sum_{k=0..2n} T(n,k)*z^k]*x^(2n)*y^n/(1-x-xy)^(4n+1) where A(x,y,x+xy) = Sum_{n>=0, k=0..n} C(n,k)^4*x^n*y^k at z = x+xy; this is the triangle of coefficients T(n,k), read by rows.

Original entry on oeis.org

1, 7, 4, 1, 131, 176, 96, 16, 1, 3067, 6588, 5895, 2416, 477, 36, 1, 79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1, 2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1, 62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1
Offset: 0

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Author

Paul D. Hanna, Mar 02 2011

Keywords

Examples

			G.f.: A(x,y,z) = 1/(1-x-x*y)
+ 2*(7 + 4*z + z^2)*x^2*y/(1-x-x*y)^5
+ 6*(131 + 176*z + 96*z^2 + 16*z^3 + z^4)*x^4*y^2/(1-x-x*y)^9
+ 20*(3067 + 6588*z + 5895*z^2 + 2416*z^3 + 477*z^4 + 36*z^5 + z^6)*x^6*y^3/(1-x-x*y)^13 +...
G.f. at z = x+xy yields: A(x,y,x+xy) = 1 + (1 + y)*x
+ (1 + 16*y + y^2)*x^2
+ (1 + 81*y + 81*y^2 + y^3)*x^3
+ (1 + 256*y + 1296*y^2 + 256*y^3 + y^4)*x^4
+ (1 + 625*y + 10000*y^2 + 10000*y^3 + 625*y^4 + y^5)*x^5 +...
which is a series involving binomial coefficients to the 4th power.
...
This triangle of coefficients T(n,k) of z^k, k=0..2n, begins:
[1];
[7, 4, 1];
[131, 176, 96, 16, 1];
[3067, 6588, 5895, 2416, 477, 36, 1];
[79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1];
[2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1];
[62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1];
[1818812387, 9876304172, 24205612067, 34939683632, 32837525567, 21029302364, 9356637759, 2899564224, 619135629, 88879924, 8237341, 461776, 14161, 196, 1];
[54257991011, 339398092544, 968547444480, 1655445817088, 1881608595776, 1496188189440, 853911382016, 353544477440, 106191762336, 22927328512, 3492995968, 364541184, 24932320, 1044736, 24192, 256, 1];
...

Crossrefs

Cf. A000897, A099601.

Formula

Row sums equal A000897(n) = (4n)!/((2n)!*n!^2).
Column 0 equals A099601(n) = quotient of de Bruijn sums S(4,n)/S(2,n).

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

Original entry on oeis.org

1, 48, 15120, 7392000, 4414410000, 2956651746048, 2133278987583744, 1621682968820428800, 1281351259836532170000, 1043032815185819858400000, 869343653096068540955685120, 738637974389826550020188712960, 637665137404661719206664998969600
Offset: 0

Views

Author

Bradley Klee, Jun 04 2019

Keywords

Comments

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.

Links

Crossrefs

Hadamard Factors: A000984, A002894, A002897, A001448, A000897, A008977.
Calabi-Yau Periods: A008978, A186420, A268553, A039699.

Programs

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

Formula

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

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

Views

Author

Ilya Gutkovskiy, Nov 18 2021

Keywords

Crossrefs

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

Programs

  • Mathematica
    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}]
  • PARI
    a(n) = (4*n)! / (n! * (2*n)!) \\ Andrew Howroyd, Nov 20 2021

Formula

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

A381199 a(n) = (4*n)!/((n!)^2*(2*n)!)*Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,k).

Original entry on oeis.org

1, 36, 6300, 1718640, 575675100, 216636756336, 87874675224336, 37563969509352000, 16692217815436148700, 7642084994921759382000, 3582530520581922083974800, 1712083670316898167464884800, 831357643152788660610464490000, 409154554816583487288034143528000, 203690783136217174743485058666840000
Offset: 0

Views

Author

Stefano Spezia, Feb 16 2025

Keywords

Links

Crossrefs

Cf. A000897, A000984, A001044, A002893, A007318, A008459, A010050, A100733.

Programs

  • Mathematica
    a[n_]:=(4n)!/((n!)^2*(2n)!)*Sum[Binomial[n,k]^2Binomial[2k,k],{k,0,n}]; Array[a,15,0]

Formula

a(n) = (4*n)!*hypergeom([1/2, -n, -n], [1, 1], 4)/((n!)^2*(2*n)!).
D-finite with recurrence n^4*a(n) -4*(4*n-1)*(4*n-3)*(10*n^2-10*n+3)*a(n-1) +144*(4*n-5)*(4*n-3)*(4*n-7)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Feb 18 2025
a(n) ~ 2^(6*n - 1/2) * 3^(2*n + 3/2) / (4*Pi^2*n^2). - Vaclav Kotesovec, May 29 2025
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