A229703 -id:A229703 - cp's OEIS frontend

A378802 a(n) = n * binomial(4*n, n).

0, 4, 56, 660, 7280, 77520, 807576, 8288280, 84146400, 847289520, 8476605280, 84362730452, 836022413616, 8255176274800, 81266247493200, 797911337890800, 7816430993273280, 76417576884236016, 745777615780501920, 7266758081613043600, 70706322844243486400, 687103929058903836480

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, A225847, A229703, A374522.

a[n_] := n * Binomial[4*n, n]; Array[a, 25, 0]

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

a(n) = n * A005810(n).
a(n) = A374522(n) + n.
a(n) == 0 (mod 4).
Sum_{n>=1} 1/a(n) = A225847.
Sum_{n>=1} (-1)^n/a(n) = A229703.

A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

Original entry on oeis.org

2, 1, 8, 3, 3, 9, 5, 4, 7, 1, 7, 7, 9, 3, 4, 4, 3, 6, 8, 7, 0, 9, 9, 8, 3, 2, 1, 0, 2, 7, 8, 8, 5, 3, 9, 1, 9, 8, 3, 0, 4, 8, 6, 4, 0, 2, 9, 2, 2, 6, 2, 2, 7, 0, 0, 1, 3, 2, 5, 6, 8, 5, 4, 9, 8, 0, 6, 6, 7, 9, 6, 6, 1, 3, 5, 9, 0, 4, 2, 7, 6, 1, 3, 1, 7, 0, 9, 3, 7, 4, 0, 2, 9, 0, 7, 9, 6, 3, 9, 3, 9, 6, 3, 3, 2

Offset: 0

Amiram Eldar, Dec 07 2024

			-0.21833954717793443687099832102788539198304864029226...

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A086465, A229703, A378806.

RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, -27/256]/ 4, 10, 120][[1]]

Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; -27/256) / 4, where 4F3 is a generalized hypergeometric function.

Equals 27*d^2/((d^2-4)*(2*d^2+1)^2) + (3*d*(d^2-1)*(2*d^2-1)/(2*(2*d^2+1)^3)) * log(abs((d-1)/(d+1))) + (3*(d^2-1)*(2*d^4-2*d^3-7*d^2-3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d+2))^(3/2) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1)) - (3*(d^2-1)*(2*d^4+2*d^3-7*d^2+3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d-2))^(3/2) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1)), where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

cp's OEIS Frontend

A378802 a(n) = n * binomial(4*n, n).

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