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A100071
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An inverse Chebyshev transform of n.
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14
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| sum{k=0..floor(n/2), binomial(n-k,k)(-1)^k*A100071(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 "An Identity Involving the Least Common Multiple of Binomial Coefficients and its Application" mentions this sequence. [From 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]
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REFERENCES
| Bakir Farhi, An Identity Involving the Least Common Multiple of Binomial Coefficients and its Application, American Mathematical Monthly, Nov. 2009, page 838 [From Roger L. Bagula, Nov 08 2009]
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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(C(n,k)*2^(n-k)*C(2k-2,k-1)(-1)^(k-1),k,0,n); - 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)!) [From Stefan Steinerberger, Nov 04 2008]
a(n) = A056040(n)*(n/2)^(n-1 mod 2). [Peter Luschny, Aug 31 2011]
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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}] [From Stefan Steinerberger, Nov 04 2008]
Table[If[n == 0, 0, n*Binomial[n - 1, Floor[(n - 1)/2]]], {n, 0, 30}] [From Roger L. Bagula, Nov 08 2009]
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CROSSREFS
| Cf. A134757.
Sequence in context: A058215 A166456 A162214 * A129912 A182863 A161507
Adjacent sequences: A100068 A100069 A100070 * A100072 A100073 A100074
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Nov 02 2004
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