A378802 -id:A378802 - cp's OEIS frontend

A378804 a(n) = n * 2^n * binomial(4*n, n).

0, 8, 224, 5280, 116480, 2480640, 51684864, 1060899840, 21541478400, 433812234240, 8680043806720, 172774871965696, 3424347806171136, 67626404043161600, 1331466198928588800, 26145958720005734400, 512257621575157678080, 10016204637370583089152, 195501127311163895316480

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36.

Cf. A005810, A036289, A378802.

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

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

a(n) = A036289(n) * A005810(n).
a(n) = 2^n * A378802(n).
a(n) == 0 (mod 8).
Sum_{n>=1} (-1)^n/a(n) = (log(2) - 6*log(3))/7 + Sum_{r: 2*r^3 + 12*r + 13 = 0} log(r+2)/(r+3) = -0.120716907732393305... (Borwein and Girgensohn, 2005, p. 32, eq. (43)).

A225847 Decimal expansion of Sum_{n>=1} 1/(nbinomial(4n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 17 2013

			0.269523929027742017317181647486329302840849825343266309815843772918628369827...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 60.

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp. 220 (2013) 331-338, Example 7.

Cf. A073010, A210453, A378802.

(1/4)*HypergeometricPFQ[{1, 1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/256] // RealDigits[#, 10, 105]& // First

Equals Integral_{x>0} ((3*x)/((1 + x)*(1 + 3*x + 6*x^2 + 4*x^3 + x^4))) dx.

Equals (3*c/(2*c^2+1)) * log((c-1)/(c+1)) + (3*(c-1)/(2*(2*c^2+1))) * sqrt(c/(c+2)) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c+1)/(2*(2*c^2+1))) * sqrt(c/(c-2)) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1)), where c = sqrt(1 + (16/sqrt(3)) * cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013). - Amiram Eldar, Dec 07 2024

A229703 Decimal expansion of Sum_{k>=1} (-1)^k/(k*binomial(4k,k)) (negated).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 27 2013

			-0.2335324174851719887871681394896038...

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp. 220 (2013) 331-338, Example 6.

Cf. A225847, A378802.

HypergeometricPFQ[{1, 1, 4/3, 5/3}, {5/4, 3/2, 7/4}, -27/256]/4 // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 18 2014 *)

Equals (3*d/(2*d^2+1))*log(abs((d-1)/(d+1))) + (3*(d-1)/(2*(2*d^2+1))) * sqrt(d/(d+2)) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1)) - (3*(d+1)/(2*(2*d^2+1))) * sqrt(d/(d-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). - Amiram Eldar, Dec 07 2024

A378803 a(n) = n^2 * binomial(4*n, n).

Original entry on oeis.org

0, 4, 112, 1980, 29120, 387600, 4845456, 58017960, 673171200, 7625605680, 84766052800, 927990034972, 10032268963392, 107317291572400, 1137727464904800, 11968670068362000, 125062895892372480, 1299098807032012272, 13423997084049034560, 138068403550647828400, 1414126456884869728000

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, A378802.

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

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

a(n) = n^2 * A005810(n).
a(n) = n * A378802(n).
a(n) == 0 (mod 4).
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 = 0.25947076781691783..., where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 336, Example 3).
Sum_{n>=1} (-1)^n/a(n) = -(3/2)*log((1-d)/(1+d))^2 + (3/4) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1))^2 + (3/4) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1))^2 = -0.24154452788843591937..., 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

A378804 a(n) = n * 2^n * binomial(4*n, n).

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A225847 Decimal expansion of Sum_{n>=1} 1/(n*binomial(4*n,n)).

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A229703 Decimal expansion of Sum_{k>=1} (-1)^k/(k*binomial(4k,k)) (negated).

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Author

Keywords

Examples

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Programs

Mathematica

Formula

A378803 a(n) = n^2 * binomial(4*n, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A225847 Decimal expansion of Sum_{n>=1} 1/(nbinomial(4n,n)).