A010050 a(n) = (2n)!.
1, 2, 24, 720, 40320, 3628800, 479001600, 87178291200, 20922789888000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 304888344611713860501504000000, 265252859812191058636308480000000
Offset: 0
Examples
G.f. = 1 + 2*x + 24*x^2 + 720*x^3 + 40320*x^4 + 3628800*x^5 + ...
References
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 110.
- H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.
- Isaac Newton, De analysi, 1669; reprinted in D. Whiteside, ed., The Mathematical Works of Isaac Newton, vol. 1, Johnson Reprint Co., 1964; see p. 20.
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 32 and 33, equations 32:6:1 and 33:6:1 at pages 300 and 314.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- W. Dunham, Touring the calculus gallery, Amer. Math. Monthly, 112 (2005), 1-19.
- Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.
- Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
- Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
- Index entries for related partition-counting sequences.
Crossrefs
Programs
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Magma
[Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Aug 21 2011
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Maple
A010050 := proc(n) (2*n)! ;end proc: # R. J. Mathar, Feb 28 2011
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Mathematica
a[n_]:=(2n)!; Array[a,16,0] (* Stefano Spezia, Jan 02 2025 *)
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PARI
a(n)=(n*2)! \\ M. F. Hasler, Apr 22 2015
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Sage
[stirling_number1(2*n+1,1) for n in range(0, 22)] # Zerinvary Lajos, Nov 26 2009
Formula
a(n) = 2^n*A000680(n).
E.g.f.: arctanh(x) = Sum_{k>=0} a(k) * x^(2*k+1)/ (2*k+1)!.
E.g.f.: 1/(1-x^2) = Sum_{k>=0} a(k) * x^(2*k) / (2*k)!. - Paul Barry, Sep 14 2004
D-finite with recurrence: a(n+1) = a(n)*(2*n+1)*(2*n+2) = a(n)*A002939(n-1). - Lekraj Beedassy, Apr 29 2005
a(n) = Product_{k = 1..n} (2*k*n-k*(k-1)). - Peter Bala, Feb 21 2011
G.f.: G(0) where G(k) = 1 + 2*x*(2*k+1)*(4*k+1)/(1 - 4*x*(k+1)*(4*k+3)/(4*x*(k+1)*(4*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2012
a(n) = 2*A002674(n), n > 0. - Wesley Ivan Hurt, Jun 05 2013
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ 2*sqrt(Pi)*4^n*n^(2*n+1/2)/exp(2*n).
Sum_{n>=0} 1/a(n) = cosh(1) = A073743. (End)
Extensions
Third line of data from M. F. Hasler, Apr 22 2015
Comments