A086467 -id:A086467 - cp's OEIS frontend

A002736 Apéry numbers: a(n) = n^2*C(2n,n).

0, 2, 24, 180, 1120, 6300, 33264, 168168, 823680, 3938220, 18475600, 85357272, 389398464, 1757701400, 7862853600, 34901442000, 153876579840, 674412197580, 2940343837200, 12759640231800, 55138611528000, 237371722628040, 1018383898440480

Offset: 0

N. J. A. Sloane

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-1 of B equals -a(n-1). - T. D. Noe, May 01 2011

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 93.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Kunle Adegoke, Robert Frontczak, and Taras Goy, Fibonacci-Catalan Series, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A110.
Paul S. Bruckman, Problem B-871, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 85.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 (Annotated scans of some selected pages).
Indulis Strazdins, Solution to problem B-871, Fibonacci Quartely, Vol. 38, No. 1 (2000), pp. 86-87.
Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, Vol. 40, No. 2 (May 2002), pp. 175-180.
Alfred J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer, Vol. 1 (1978/1979), pp. 195-203.

Cf. A000108, A002544, A005258, A005259, A005429, A005430, A086467.

A diagonal of A331431.

[n^2*Binomial(2*n, n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

seq(n^2*binomial(2*n,n), n=0..50); # Robert Israel, Aug 07 2014

CoefficientList[ Series[x (4 x + 2)/(1 - 4 x)^(5/2), {x, 0, 20}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[n^2 Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Jun 21 2017 *)

combinat::catalan(n)*(n+1)*n^2 $ n = 0..36 // Zerinvary Lajos, Apr 17 2007

my(x='x+O('x^100)); concat(0, Vec(x*(4*x+2)/((1-4*x)^(5/2)))) \\ Altug Alkan, Mar 21 2016

a(n) = n^2*binomial(2*n, n); \\ Michel Marcus, Mar 21 2016

[n^2*(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2022

G.f.: x*(4*x+2)/((1-4*x)^(5/2)). - Marco A. Cisneros Guevara, Jul 25 2011
Sum_{n>=1} 1/a(n) = Pi^2/18 (Euler). - Benoit Cloitre, Apr 07 2002
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 4^n*n^(3/2)/sqrt(Pi).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)^2 = A086467, where phi is the golden ratio. (End)
D-finite with recurrence: (-n+1)*a(n) +2*(n+4)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 21 2020
a(n) = (2n)!/(Gamma(n))^2. - Diego Rattaggi, Mar 30 2020
a(n) = Sum_{k=0..2*n} binomial(2*n,k)*abs(n-k)^3 (Bruckman, 1999; Strazdins, 2000). - Amiram Eldar, Jan 12 2022
Sum_{n>=1} x^n/a(n) = 2*arcsin(sqrt(x)/2)^2, for abs(x) < 4 (Adegoke et al., 2022, section 5, p. 10). - Amiram Eldar, Dec 07 2024
From Peter Bala, Aug 02 2025: (Start)
For n >= 1,
a(n) = 2*n*(2*n-1)/(n-1)^2 * a(n-1) with a(1) = 2 and
1/a(n) = Sum_{k = 0..n} (-1)^(n+k+1) * binomial(n, k)*binomial(n+k, k)/(n+k)^2. (End)
a(n) = 2 * A002544(n-1) for n>=1. - Alois P. Heinz, Aug 03 2025

A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

Equals the value of the Dirichlet L-series of the non-principal character modulo 5 (A080891) at s=1. - Jianing Song, Nov 16 2019

			0.43040894096400403888943323295060542542457...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.

R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22 for L(m=5,r=3,s=1).
H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers

Cf. A086465, A086467, A086468.

Cf. A000032, A000045, A080891, A328717 (s=2), A344041.

2*Log[GoldenRatio]/Sqrt[5] // RealDigits[#, 10, 102]& // First (* Jean-François Alcover, Apr 18 2014 *)

2*log((1+sqrt(5))/2)/sqrt(5) \\ Stefano Spezia, Oct 15 2024

Equals Sum_{k>=1} (-1)^(k-1)/(k*binomial(2*k,k)).
Equals A010532 * A002390 / 10. - R. J. Mathar, Jul 26 2010
Also equals f'(0) = 2*log(phi)/sqrt(5), with f(x) = (phi^x-cos(Pi*x)*phi^-x)/sqrt(5), the real Fibonacci interpolating function. - Jean-François Alcover, Apr 04 2014
Equals Sum_{k>=1} A080891(k)/k = Sum_{k>=1} Kronecker(5,k)/k = 1 - 1/2 - 1/3 + 1/4 + 1/6 - 1/7 - 1/8 + 1/9 + ... - Jianing Song, Nov 16 2019
Equals Sum_{k>=1} F(k)/(k*2^(k+1)), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Aug 10 2020
Sum_{k>=1} (2*k+1)*Lucas(k)/(k*(k+1)*2^k) = 10*c + 2 = 6.3040894096... where c is this constant (Seiffert, 1994). - Amiram Eldar, Jan 15 2022
Equals Sum_{k>=1} F(k)/(k*3^k), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Jul 02 2023
Equals 1/Product_{p prime} (1 - Kronecker(5,p)/p), where Kronecker(5,p) = 0 if p = 5, 1 if p == 1 or 4 (mod 5) or -1 if p == 2 or 3 (mod 5). - Amiram Eldar, Dec 17 2023
Equals A344041/2. - Hugo Pfoertner, Oct 16 2024

A086465 Decimal expansion of (5 + 4sqrt(5)arcsch(2))/25.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

			0.37216357638560161555577...

D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly 92, No. 7 (1985) 449-457.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Wikipedia, Inverse hyperbolic function: Series expansions.
Index entries for transcendental numbers.

Cf. A086466, A086467, A086468.

2/625*(14*sqrt(5)*log((1+sqrt(5))/2)+5) ; # R. J. Mathar, Mar 04 2009

RealDigits[(5 + 4*Sqrt[5]*ArcSinh[1/2])/25, 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

suminf(n=1, (-1)^(n-1)/binomial(2*n,n)) \\ Michel Marcus, Jul 31 2015

asinh(.5)*sqrt(5)*.16+.2 \\ Use \p99 to get 99 digits. - M. F. Hasler, Jul 31 2015

Equals Sum_{n>=1} (-1)^(n-1)/binomial(2*n,n).

Corrected definition and digits by a factor of 25/24. - R. J. Mathar, Mar 04 2009

A086468 Decimal expansion of 2*zeta(3)/5.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 21 2003

			0.48082276126383771415989526460457999630599451693620...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.46.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 446.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.

Cf. A002117, A086465, A086466, A086467.

SetDefaultRealField(RealField(250)); L:=RiemannZeta();  2*Evaluate(L,3)/5; // G. C. Greubel, Nov 02 2018

First[RealDigits[N[2*Zeta[3]/5, 100]]] (* Stefano Spezia, Nov 02 2018 *)

2*zeta(3)/5 \\ Michel Marcus, Nov 02 2018

Equals Sum_{n>=1} (-1)^(n-1)/(n^3*binomial(2*n,n)).
Equals 2*A002117/5. - R. J. Mathar, Feb 08 2009
Equals (1/10)*Sum_{k>=1} (30*k - 11)/((2*k - 1)*k^3*binomial(2*k,k)^2) (see Finch). - Stefano Spezia, Nov 01 2024

A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

Original entry on oeis.org

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

Offset: 0

Robert Bilinski, Dec 06 2020

			0.2315648205771943924969290712315327600164063500492988708153012286...

Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, 2000.
Index entries for transcendental numbers

Cf. A002390, A001622, A086467.

RealDigits[Log[GoldenRatio]^2, 10, 100][[1]] (* Amiram Eldar, Dec 06 2020 *)

asinh(1/2)^2 \\ Michel Marcus, Dec 06 2020

Equals arcsinh(1/2)^2 = A002390^2.
Equals (1/2)*Sum_{k>=1} ((k!)^2*(-1)^(k+1))/((2*k)!*k^2) = A086467/2.
Equals (1/3)*(zeta(2) - Sum_{k>=1} ((k!)^2*(-1)^k)/((2*k)!*(2*k+1)^2)).
Equals (1/2)*Sum_{k>=1} (-1)^(k+1)/A002736(k).

cp's OEIS Frontend

A002736 Apéry numbers: a(n) = n^2*C(2n,n).

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A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).

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A086465 Decimal expansion of (5 + 4*sqrt(5)*arcsch(2))/25.

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A086468 Decimal expansion of 2*zeta(3)/5.

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A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

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A086465 Decimal expansion of (5 + 4sqrt(5)arcsch(2))/25.