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A100071 a(n) = n * binomial(n-1, floor((n-1)/2)) = n * max_{i=0..n} binomial(n-1, i). 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Old name: An inverse Chebyshev transform of n.

Sum_{k = 0..floor(n/2)} binomial(n-k,k)*(-1)^k*a(n-2k) = 1.

Hankel transform is (-1)^n*n*2^(n-1), A085750. This is the inverse binomial transform of -n. - Paul Barry, Jan 11 2007

Corollary 3 of the Farhi reference mentions this sequence. - Roger L. Bagula, Nov 08 2009

Number of UDUD's in all length n+3 left factors of Dyck paths (here U=(1,1) and D=(1,-1)). Example: a(2)=2 because in (UDUD)U, UDUUD, UDUUU, UUDDU, U(UDUD), UUDUU, UUUDD, UUUDU, UUUUD, and UUUUU we have a total of two UDUDs (shown between parentheses).  Also number of UUDD's in all length n+3 left factors of Dyck paths (here U=(1,1) and D=(1,-1)). Example: a(2)=2 because in UDUDU, UDUUD, UDUUU, (UUDD)U, UUDUD, UUDUU, U(UUDD), UUUDU, UUUUD, and UUUUU we have a total of two UUDDs (shown between parentheses). - Emeric Deutsch, Jun 19 2011

Apparently the number of long ascents in all symmetric Dyck (n+1)-paths. - David Scambler, Aug 17 2012

Beginning with the least positive term multiple of an odd prime p (which is a(p)), we have exactly p+1 consecutive terms multiple of p. - Vladimir Shevelev, Aug 17 2012

Apparently also the count of 'unmatched symbols' in the binary strings of length n (see A008314). - Wouter Meeussen, May 26 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Ruggero Bandiera, Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. See p. 34.

F. Disanto, A. Frosini, S. Rinaldi, Square involutions, J. Int. Seq. 14 (2011) # 11.3.5

Bakir Farhi, An Identity Involving the Least Common Multiple of Binomial Coefficients and its Application, American Mathematical Monthly, Nov. 2009, page 838.

Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).

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).

a(n) = n*C(n-1,floor((n-1)/2)); a(n) = Sum_{k = 0..n} C(n,k)*2^(n-k)*C(2*k - 2,k - 1)(-1)^(k-1). - Paul Barry, Jan 11 2007

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 Banderia 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

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

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 *)

PROG

(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

(MAGMA) [n*Binomial(n-1, Floor((n-1)/2)): n in [0..35]]; // Vincenzo Librandi, Sep 14 2015

(PARI) a(n) = n * binomial(n-1, (n-1)\2); \\ Michel Marcus, Sep 14 2015

CROSSREFS

Cf. A134757, A008314.

Sequence in context: A166456 A162214 A309728 * A331552 A129912 A283477

Adjacent sequences:  A100068 A100069 A100070 * A100072 A100073 A100074

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 02 2004

EXTENSIONS

Name changed, using part of a comment from Paul Barry, by Peter Luschny, Aug 17 2012

STATUS

approved

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Last modified February 25 04:03 EST 2021. Contains 341596 sequences. (Running on oeis4.)