|
| |
|
|
A001448
|
|
C(4n,2n) = (4*n)!/((2*n)!*(2*n)!).
|
|
9
|
|
|
|
1, 6, 70, 924, 12870, 184756, 2704156, 40116600, 601080390, 9075135300, 137846528820, 2104098963720, 32247603683100, 495918532948104, 7648690600760440, 118264581564861424, 1832624140942590534
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Corollary 8 in Chapman et alia says: "For n>=1, there are binomial(4n,2n) binary sequences of length 4n+1 with the property that for all j, the jth occurrence of 10 appears in positions 4j+1 and 4j+2 or later (if it exists at all)." - Peter Luschny, Nov 21 2011
|
|
|
REFERENCES
|
R. J. Chapman, T. Y. Chowa, A. Khetana, D. P. Moulton and R. J. Waters, Simple formulas for lattice paths avoiding certain periodic staircase boundaries, Journal of Combinatorial Theory, Series A 116 (2009) 205-214.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 0..100
|
|
|
FORMULA
|
Using Stirling's formula in sequence A000142 it is easy to get the asymptotic expression a(n) ~ 16^n / sqrt(2 * Pi * n) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
a(n)= 2*A001700(2*n-1) = (2*n+1)*C(2*n), n >= 1, C(n) := A000108(n) (Catalan). G.f.: (1-y*((1+4*y)*c(y)-(1-4*y)*c(-y)))/(1-(4*y)^2) with y^2=x, c(y)= g.f. for A000108 (Catalan). - Wolfdieter Lang, Dec 13 2001
a(n) ~ 2^(-1/2)*pi^(-1/2)*n^(-1/2)*2^(4*n)*{1 - 1/16*n^-1 + ...} - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
a(n) = (1/Pi)*integral(x=-2..2, (2+x)^(2*n)/sqrt((2-x)*(2+x))). Peter Luschny, Sep 12 2011]
G.f.: (1/2) * (1/(1+4*x^(1/2))^(1/2) + 1/(1-4*x^(1/2))^(1/2)) - Mark van Hoeij, Oct 25 2011.
sum_{n>=1} 1/a(n) = 16/15 +pi*sqrt(3)/27 -2*sqrt(5)*log(phi)/25, [T. Trif, Fib Quart 38 (2000) 79] with phi=A001622. - R. J. Mathar, Jul 18 2012
n*(2*n-1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Dec 02 2012
|
|
|
EXAMPLE
|
a(n)=(1/pi)*int(x^(2n)/sqrt(4-(x-2)^2),x,0,4). [From Paul Barry, Sep 17 2010]
|
|
|
MAPLE
|
A001448 := n-> binomial(4*n, 2*n) ;
|
|
|
PROG
|
(MAGMA) [Factorial(4*n)/(Factorial(2*n)*Factorial(2*n)): n in [0..20]]; // Vincenzo Librandi, Sep 13 2011
(PARI) a(n)=binomial(4*n, 2*n) \\ Charles R Greathouse IV, Sep 13 2011
|
|
|
CROSSREFS
|
Bisection of A000984. Cf. A002458.
Sequence in context: A048708 A104900 A186667 * A024489 A036361 A182563
Adjacent sequences: A001445 A001446 A001447 * A001449 A001450 A001451
|
|
|
KEYWORD
|
nonn,nice,easy,changed
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
