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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A005810 a(n) = binomial(4n,n).

Original entry on oeis.org

1, 4, 28, 220, 1820, 15504, 134596, 1184040, 10518300, 94143280, 847660528, 7669339132, 69668534468, 635013559600, 5804731963800, 53194089192720, 488526937079580, 4495151581425648, 41432089765583440, 382460951663844400
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Start off with 0 balls in a box. Find the number of ways you can throw 3 balls back out. Then continue to throw 4 balls into the box after each stage. (I.e., the first stage is 0. Then at the next stage there are 4 ways to throw 3 balls back out.) - Ruppi Rana (ruppirana007(AT)hotmail.com), Mar 03 2004
Central coefficients of A094527. - Paul Barry, Mar 08 2011
This is the case m = 2n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. - Tom Copeland, Oct 10 2012
Conjecture: a(n) == 4 (mod n^3) iff n is prime. - Gary Detlefs, Apr 03 2013
For prime p, the congruence a(p) = binomial(4*p,p) = 4 (mod p^3) is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014

Examples

			G.f. = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 + 134596*x^6 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row 4 of A060539.
binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
Cf. A007318, A182400, A262261, A378806, A378807.

Programs

Formula

a(n) is asymptotic to c*(256/27)^n/sqrt(n) with c = sqrt(2 / (3 Pi)) = 0.460658865961780639... - Benoit Cloitre, Jan 26 2003; corrected by Charles R Greathouse IV, Dec 14 2006
a(n) = Sum_{k=0..2n} binomial(2n,k) * binomial(2n,k-n). - Paul Barry, Mar 08 2011
G.f.: g/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
D-finite with recurrence: 3*n*(3*n-1)*(3*n-2)*a(n) - 8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 02 2012
a(n) = binomial(4*n,n-1)*(3*n+1)/n. - Gary Detlefs, Apr 03 2013
a(n) = C(4*n-1,n-1)*C(16*n^2,2)/(3*n*C(4*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
a(n) = Sum_{i,j,k = 0..n} binomial(n,i)*binomial(n,j)*binomial(n,k)* binomial(n,i+j+k). - Peter Bala, Dec 28 2014
a(n) = GegenbauerC(n, -2*n, -1). - Peter Luschny, May 07 2016
From Ilya Gutkovskiy, Nov 22 2016: (Start)
O.g.f.: 3F2(1/4,1/2,3/4; 1/3,2/3; 256*x/27).
E.g.f.: 3F3(1/4,1/2,3/4; 1/3,2/3,1; 256*x/27). (End)
a(n) = hypergeom([-3*n, -1*n], [1], 1). - Peter Luschny, Mar 19 2018
RHS of the identity Sum_{k = 0..2*n} (-1)^(n+k)*binomial(4*n, k)* binomial(4*n, 2*n-k) = binomial(4*n,n). - Peter Bala, Oct 07 2021
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) satisfies the differential equation
(-256*x^3 + 27*x^2)*A(x)''' + (-1152*x^2 + 54*x)*A(x)'' + (-816*x + 6)*A(x)' - 24*A(x) = 0 with A(0) = 1, A'(0) = 4 and A''(0) = 56.
Algebraic equation: (1 - A(x))*(1 + 3*A(x))^3 + 256*x*A(x)^4 = 0.
Sum_{n >= 1} a(n)*( x*(3*x + 4)^3/(256*(1 + x)^4) )^n = x. (End)
From Amiram Eldar, Dec 07 2024: (Start)
Sum_{n>=1} 1/a(n) = A378806.
Sum_{n>=1} (-1)^n/a(n) = A378807. (End)
From Peter Bala, Jun 29 2025: (Start)
a(n) = (1/8)^n * Sum_{k = n..4*n} binomial(k, n) * binomial(4*n, k).
Sum_{n >= 0 } a(n)*(1/128)^n = (1/5)*(sqrt(2) + sqrt(7 + 5*sqrt(2))). (End)
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+1,k).
G.f.: 1/(1 - 4*x*g^3) where g = 1+x*g^4 is the g.f. of A002293. (End)

Extensions

More terms from Henry Bottomley, Oct 06 2000
Corrected by T. D. Noe, Jan 16 2007

A378806 Decimal expansion of Sum_{k>=1} 1/binomial(4*k, k).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Dec 07 2024

Keywords

Examples

			0.29088207152128721276259725668681035773368176167609...

Links

Crossrefs

Cf. A005810, A073016, A229705, A378807.

Programs

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

Formula

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*c^2/((c^2-4)*(2*c^2+1)^2) + (3*c*(c^2-1)*(2*c^2-1)/(2*(2*c^2+1)^3)) * log((c-1)/(c+1)) + (3*(c^2-1)*(2*c^4-2*c^3-7*c^2-3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c+2))^(3/2) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c^2-1)*(2*c^4+2*c^3-7*c^2+3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c-2))^(3/2) * 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, p. 337, Example 9).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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