A100734 -id:A100734 - cp's OEIS frontend

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).

1, 122, 114126, 169305620, 307902541870, 628881704226972, 1384648756554128604, 3213280613371692112392, 7752574653184355259506670, 19272593072633780827550508620, 49062146831202726778631520779476, 127331178560917294198014376933764792, 335791906923524740189894975371277920796

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 128.

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. 16.

Cf. A000442, A000984, A007318, A010050, A100734.

Cf. A381161, A381162, A381163, A381164.

a[n_]:=Sum[Binomial[2n,n]Binomial[n, k](5k)!/((k!)^3*(2k)!), {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-4*x))/sqrt(1-4*x).
a(n) = binomial(2*n,n)*hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1/2, 1, 1, 1], -5^5/4).
a(n) ~ 3^(n + 3/2) * 7^(n + 3/2) * 149^(n +3/2) / (4 * 5^7 * 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)

cp's OEIS Frontend

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).

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

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Magma

Mathematica

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

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Magma

Mathematica

Formula

cp's OEIS Frontend

A381165 a(n) = Sum_{k=0..n} binomial(2*n,n)*binomial(n, k)*(5*k)!/((k!)^3*(2*k)!).

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Author

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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

A381165 a(n) = Sum_{k=0..n} binomial(2n,n)binomial(n, k)(5k)!/((k!)^3(2k)!).