A210453 -id:A210453 - cp's OEIS frontend

A090763 a(n) = (3n+3)!/(3n!(2n+2)!).

1, 10, 84, 660, 5005, 37128, 271320, 1961256, 14060475, 100150050, 709634640, 5006710800, 35197176924, 246681069040, 1724337127920, 12025860872784, 83702724824775, 581558091471630, 4034231805704100, 27945630038703300

Offset: 0

Al Hakanson (hawkuu(AT)excite.com), Feb 15 2004

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A005809, A025174, A210453, A360560.

a:= n->sum(j*binomial(n+2, j)*binomial(2*(n+1), j)/6, j=0..n+2): seq(a(n), n=0..21); # Zerinvary Lajos, Jul 31 2006
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      3*(3*n+1)*(3*n+2)*a(n-1)/(2*n*(2*n+1)))
    end:
seq(a(n), n=0..30); # Alois P. Heinz, Feb 01 2014

a[n_] := 1/Integrate[(x^(2/3) - x)^n, {x, 0, 1}]; Table[ a[n], {n, 0, 19}] (* Robert G. Wilson v, Feb 18 2004 *)
a[n_] := 1/(2*Beta[2n, n]) (* Enrique Pérez Herrero, May 17 2009 *)
a[n_] : =1/2*Sum[j*Binomial[2 n, j]*Binomial[n, j], {j, 1, n}] (* Enrique Pérez Herrero, May 22 2009 *)

[binomial(3*n,n)*n/3 for n in range(1,21)] # Zerinvary Lajos, May 17 2009

a(n) = 1/(Integral_{x=0..1} (x^(2/3)-x)^n dx).
a(n) = 1/(Integral_{x=0..1} (x-x^1.5)^n dx).
a(n) = 1/(2*Beta(2n,n)). - Enrique Pérez Herrero, May 17 2009
a(1) = 1; a(n) = a(n-1)*2*binomial(3n,3)/binomial(2n,3). - Enrique Pérez Herrero, May 19 2009
a(n) = (1/2)*Sum{j=1,n}(j*binomial(2n,j)*binomial(n,j)). - Enrique Pérez Herrero, May 22 2009
a(n) = (n+1)*A025174(n+1). - R. J. Mathar, Jun 21 2009
G.f.: Hypergeometric2F1(4/3, 5/3, 3/2, 27*x/4). - Stefano Spezia, Oct 18 2019
G.f.: (-(3*sqrt(4-27*x)*csc(arcsin((3*sqrt(3*x))/2)/3)^2)/((4*(4-27*x)^(3/2)))+(sqrt(3)*cot(arcsin((3*sqrt(3*x))/2)/3))/((4-27*x)*sqrt(x)*sqrt(4-27*x))). - Vladimir Kruchinin, Feb 12 2023
From Amiram Eldar, Dec 07 2024: (Start)
a(n) = (n+1) * A005809(n+1) / 3.
Sum_{n>=0} 1/a(n) = 3 * A210453. (End)

More terms from Robert G. Wilson v, Feb 18 2004

Simpler description from Vladeta Jovovic, Feb 22 2004

A144485 a(n) = (3n + 2)*binomial(3n + 1,n).

Original entry on oeis.org

2, 20, 168, 1320, 10010, 74256, 542640, 3922512, 28120950, 200300100, 1419269280, 10013421600, 70394353848, 493362138080, 3448674255840, 24051721745568, 167405449649550, 1163116182943260, 8068463611408200, 55891260077406600

Offset: 0

Roger L. Bagula, Oct 12 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Ömer Eğecioğlu, Timothy Redmond, and Charles Ryavec, Almost product evaluation of Hankel Determinants, The Electronic Journal of Combinatorics, Vol. 15, No. 1 (2008), #R6; arXiv preprint, arXiv:0704.3398 [math.CO], 2007.

Cf. A005809, A045721, A090763, A210453.

[(3*n+2)*Binomial(3*n+1, n): n in [0..20]]; // Vincenzo Librandi, Feb 14 2014

a:= proc(n) option remember; `if`(n=0, 2,
      3*(3*n+1)*(3*n+2)*a(n-1)/(2*n*(2*n+1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 01 2014

a[k_] = (3k + 2)Binomial[3k + 1, k]; Table[a[k], {k, 0, 30}]

a(n) = (3n+2)*A045721(n). - R. J. Mathar, Feb 01 2014
a(n) = 2*A090763(n). - Alois P. Heinz, Feb 01 2014
From Amiram Eldar, Dec 07 2024: (Start)
a(n) = 2 * (n+1) * A005809(n+1) / 3.
Sum_{n>=0} 1/a(n) = (3/2) * A210453. (End)

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

A225848 Decimal expansion of sum_{n>=1} 1/(nbinomial(5n,n)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 17 2013

Equals Integral_{x>0}((4*x)/((1 + x)*(1 + 4*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5))).

			0.211899379477988040620147684227922292577929674144068015360748567774676347301...

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

Cf. A073010, A210453.

(1/5)*HypergeometricPFQ[{1, 1, 5/4, 3/2, 7/4}, {6/5, 7/5, 8/5, 9/5}, 256/3125] // RealDigits[#, 10, 105]& // First

cp's OEIS Frontend

A090763 a(n) = (3*n+3)!/(3*n!*(2*n+2)!).

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A144485 a(n) = (3n + 2)*binomial(3n + 1,n).

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

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A225848 Decimal expansion of sum_{n>=1} 1/(n*binomial(5*n,n)).

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A090763 a(n) = (3n+3)!/(3n!(2n+2)!).

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

A225848 Decimal expansion of sum_{n>=1} 1/(nbinomial(5n,n)).