A100733 -id:A100733 - cp's OEIS frontend

A381163 a(n) = Sum_{k=0..n} binomial(n,k)(4k)!(2k)!/(k!)^6.

1, 49, 15217, 7437505, 4444068913, 2978797867489, 2151085262277121, 1636678166183569873, 1294384621280668799665, 1054623536679756097536097, 879831837105310233485202337, 748258333337818719124808979313, 646586399881218539235007860940609, 566284969531710881501724274920081265

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 76.

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. 15-16.

Cf. A007318, A010050, A100733, A307618.

Cf. A381161, A381162, A381164, A381165.

a[n_]:=Sum[Binomial[n,k](4k)!(2k)!/k!^6,{k,0,n}]; Array[a,14,0]

G.f.: hypergeom([1/2, 1/2, 1/4, 3/4], [1, 1, 1], 2^10*x/(1-x))/(1-x).
a(n) = hypergeom([1/4, 1/2, 1/2, 3/4, -n], [1, 1, 1, 1], -2^10).
a(n) == 1 (mod 48).
a(n) ~ 5^(2*n+4) * 41^(n+2) / (2^(41/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

A195393 a(n) = (9*n)!.

Original entry on oeis.org

1, 362880, 6402373705728000, 10888869450418352160768000000, 371993326789901217467999448150835200000000, 119622220865480194561963161495657715064383733760000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391, A195392.

Magma
```
[Factorial(9*n): n in [0..10]];
```

(9Range[0,10])! (* Harvey P. Dale, Jan 25 2023 *)

A195394 a(n) = (10*n)!

Original entry on oeis.org

1, 3628800, 2432902008176640000, 265252859812191058636308480000000, 815915283247897734345611269596115894272000000000, 30414093201713378043612608166064768844377641568960512000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Cf. A000142, A008592, A010050, A100732, A100733, A100734, A195390, A195391, A195392, A195393.

Cf. A195070, A196498.

Magma
```
[Factorial(10*n): n in [0..10]];
```

a[n_] := (10*n)!; Array[a, 6, 0] (* Amiram Eldar, Apr 03 2021 *)
(10*Range[0,10])! (* Harvey P. Dale, Aug 04 2025 *)

From Amiram Eldar, Apr 03 2021: (Start)
a(n) = A000142(A008592(n)).
Sum_{n>=0} 1/a(n) = A195070.
Sum_{n>=0} (-1)^n/a(n) = A196498. (End)

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

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 16 2025

nonn

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 p. 24.

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

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

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

cp's OEIS Frontend

A381163 a(n) = Sum_{k=0..n} binomial(n,k)(4k)!(2k)!/(k!)^6.

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A195393 a(n) = (9*n)!.

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Magma

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A195394 a(n) = (10*n)!

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Magma

Mathematica

Formula

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

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cp's OEIS Frontend

A381163 a(n) = Sum_{k=0..n} binomial(n,k)*(4*k)!*(2*k)!/(k!)^6.

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Formula

A195393 a(n) = (9*n)!.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A195394 a(n) = (10*n)!

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

Formula

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

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Mathematica

Formula

A381163 a(n) = Sum_{k=0..n} binomial(n,k)(4k)!(2k)!/(k!)^6.

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