A378804 -id:A378804 - cp's OEIS frontend

A386918 a(n) = 2^n * binomial(4*n,n).

1, 8, 112, 1760, 29120, 496128, 8614144, 151557120, 2692684800, 48201359360, 868004380672, 15706806542336, 285362317180928, 5202031080243200, 95104728494899200, 1743063914667048960, 32016101348447354880, 589188508080622534656, 10861173739509105295360

Offset: 0

Seiichi Manyama, Aug 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A066381, A386919, A386920.

Cf. A005810, A059304, A228484, A378804.

[2^n * Binomial(4*n,n): n in [0..26]]; // Vincenzo Librandi, Aug 11 2025

Table[2^n*Binomial[4*n,n],{n,0,30}] (* Vincenzo Librandi, Aug 11 2025 *)

PARI
```
a(n) = 2^n*binomial(4*n, n);
```

a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(4*n)/(1-x)^(3*n+1).
a(n) = [x^n] 1/(1-2*x)^(3*n+1).
a(n) = [x^n] (1+2*x)^(4*n).

A378805 a(n) = n^2 * 2^n * binomial(4*n, n).

Original entry on oeis.org

0, 8, 448, 15840, 465920, 12403200, 310109184, 7426298880, 172331827200, 3904310108160, 86800438067200, 1900523591622656, 41092173674053632, 879143252561100800, 18640526785000243200, 392189380800086016000, 8196121945202522849280, 170275478835299912515584, 3519020291600950115696640

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A007758, A378803, A378804.

a[n_] := n^2 * 2^n * Binomial[4*n, n]; Array[a, 20, 0]

PARI
```
a(n) = n^2 * 2^n * binomial(4*n, n);
```

a(n) = A007758(n) * A005810(n).
a(n) = 2^n * A378803(n).
a(n) = n * A378804(n).
a(n) == 0 (mod 8).
Sum_{n>=1} 1/a(n) = -(3/2)*log((c-1)/(c+1))^2 + (3/4) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1))^2 + (3/4) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1))^2 = 0.12729750445123620540..., where c = sqrt(1 + (16*sqrt(2/3))*cos(arctan(sqrt(485/27))/3)) (Batir and Sofo, 2013, p. 336, Example 1).

cp's OEIS Frontend

A386918 a(n) = 2^n * binomial(4*n,n).

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