A100071 a(n) = n * binomial(n-1, floor((n-1)/2)) = n * max_{i=0..n} binomial(n-1, i).
0, 1, 2, 6, 12, 30, 60, 140, 280, 630, 1260, 2772, 5544, 12012, 24024, 51480, 102960, 218790, 437580, 923780, 1847560, 3879876, 7759752, 16224936, 32449872, 67603900, 135207800, 280816200, 561632400, 1163381400, 2326762800
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Ruggero Bandiera and Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. See p. 34.
- F. Disanto, A. Frosini, and S. Rinaldi, Square involutions, J. Int. Seq., Vol. 14 (2011), Article 11.3.5.
- Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, The American Mathematical Monthly, Vol. 116, No. 9 (2009), pp. 836-839, see p. 838; arXiv preprint, arXiv:0906.2295 [math.NT], 2009.
- Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
- D. Nečas, Elementary models of low-pressure plasma polymerisation into nanofibrous mats, Phys. Scr. (2025) Vol. 100, 055601. See p. 10.
- Helmut Prodinger, Dispersed Dyck paths revisited, arXiv:2402.13026 [math.CO], 2024.
Programs
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Magma
[n*Binomial(n-1, Floor((n-1)/2)): n in [0..35]]; // Vincenzo Librandi, Sep 14 2015
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Maple
swing := n -> n!/iquo(n,2)!^2: A100071 := n -> swing(n)*(n/2)^(n-1 mod 2): seq(A100071(i),i=0..30); # Peter Luschny, Aug 31 2011
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Mathematica
Table[(Floor[n/2] + Ceiling[n/2] + 1)!/(Floor[n/2]!*Ceiling[n/2]!), {n, 1, 40}] (* Stefan Steinerberger, Nov 04 2008 *) Table[If[n == 0, 0, n*Binomial[n - 1, Floor[(n - 1)/2]]], {n, 0, 30}] (* Roger L. Bagula, Nov 08 2009 *); Table[ Tr[ Table[Count[match[-1 + 2*IntegerDigits[n, 2, k]], 0], {n, 2^(k - 1), 2^k - 1}]], {k, 16}] (* function 'match' see A008314; Wouter Meeussen, May 26 2013 *) -
PARI
a(n) = n * binomial(n-1, (n-1)\2); \\ Michel Marcus, Sep 14 2015
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Sage
def A100071(n): f = factorial(n)/factorial(n//2)^2 return f if is_odd(n) else f*(n/2) [A100071(n) for n in (0..50)] # Peter Luschny, Aug 17 2012
Formula
G.f.: 2*x*(1 - sqrt(1 - 4*x^2))/(sqrt(1 - 4*x^2)*(sqrt(1 - 4*x^2) + 2*x - 1)^2).
G.f.: (1/sqrt(1 - 4*x^2))*x*c(x^2)/(1 - x*c(x^2))^2.
a(n) = Sum_{k = 0..floor(n/2)} binomial(n,k)*(n - 2*k).
Sum_{k = 0..floor(n/2)} binomial(n-k,k)*(-1)^k*a(n-2k) = 1.
From Paul Barry, Jan 11 2007: (Start)
a(n) = n*binomial(n-1, floor((n-1)/2));
a(n) = Sum_{k = 0..n} binomial(n,k)*2^(n-k)*binomial(2*k-2, k-1)*(-1)^(k-1). (End)
Starting (1, 2, 6, 12, ...), = inverse binomial transform of A134757: (1, 3, 11, 37, 123, 401, ...). - Gary W. Adamson, Nov 08 2007
a(n) = a(n-1)*n/floor(n/2) for n > 0. - Reinhard Zumkeller, Jan 20 2008
G.f.: x/((1 - 2*x)*sqrt(1 - 4*x^2)). - Paul Barry, Apr 25 2008
a(n) = (floor(n/2) + ceiling(n/2) + 1)!/(floor(n/2)! * ceiling(n/2)!). - Stefan Steinerberger, Nov 04 2008
a(n) = A056040(n)*(n/2)^((n-1) mod 2). - Peter Luschny, Aug 31 2011
Asymptotic: a(n) ~ b(n) where b(n) = ceiling(2^(n-1)*sqrt(2*n-(-1)^n)/sqrt(Pi)). b(n) is also a lower bound of a(n) and an upper bound of 2^(n-1). With corollary 3 from Bakir Farhi (see reference) lcm(1,2,...,n) >= a(n) >= b(n) >= 2^(n-1). - Peter Luschny, Aug 17 2012
a(n) = n for n < 3, a(n) = 4*a(n-2) + 2*a(n-1)/(n-1) for n >= 3. - Alexander R. Povolotsky, Aug 17 2012
E.g.f.: x*(BesselI(0,2*x) + BesselI(1,2*x)). - Peter Luschny, Aug 19 2012
a(n) = (-1)^(n*(n+1)/2) * Sum_{k = 0..n} (-1)^k*k*binomial(n,k)^2. - Peter Bala, Jul 25 2016
a(n) = n!/(floor((n-1)/2)!*ceiling((n-1)/2)!). See the Bandiera link. - Michel Marcus, Feb 28 2017
D-finite with recurrence (-n+1)*a(n) + 2*a(n-1) + 4*(n-1)*a(n-2) = 0. - R. J. Mathar, Aug 09 2017
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)). (End)
Extensions
Name changed, using part of a comment from Paul Barry, by Peter Luschny, Aug 17 2012
Comments