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A001448 a(n) = C(4n,2n) or (4*n)!/((2*n)!*(2*n)!). 21
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 j-th occurrence of 10 appears in positions 4j+1 and 4j+2 or later (if it exists at all)." - Peter Luschny, Nov 21 2011

Sequence terms are given by [x^n] ( (1 + x)^(k+2)/(1 - x)^k )^n for k = 2. See the cross references for related sequences obtained from other values of k . - Peter Bala, Sep 29 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

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.

M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.

Ricardo A. Podestá, New identities for binary Krawtchouk polynomials, binomial coefficients and Catalan numbers, arXiv:1603.09156 [math.CO], 2016.

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

G.f.:  sqrt((1 + sqrt(1-16*x))/(2*(1-16*x))) = 1 + 6*x/(G(0)-6*x), where G(k)= 2*x*(4*k+3)*(4*k+1) + (2*k+1)*(k+1) - 2*x*(k+1)*(2*k+1)*(4*k+5)*(4*k+7)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

a(n) = hypergeom([1-2*n,-2*n],[2],1)*(2*n+1). - Peter Luschny, Sep 22 2014

0 = a(n)*(+65536*a(n+2) - 16896*a(n+3) + 858*a(n+4)) + a(n+1)*(-3584*a(n+2) + 1176*a(n+3) - 66*a(n+4)) + a(n+2)*(+14*a(n+2) - 14*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Oct 22 2014

0 = a(n)^2*(+196608*a(n+1)^2 - 40960*a(n+1)*a(n+2) + 2100*a(n+2)^2) + a(n)*a(n+1)*(-12288*a(n+1)^2 + 2840*a(n+1)*a(n+2) - 160*a(n+2)^2) + a(n+1)^2*(+180*a(n+1)^2 - 48*a(n+1)*a(n+2) + 3*a(n+2)^2) for all n in Z. - Michael Somos, Oct 22 2014

a(n) = [x^n] ( (1 + x)^4/(1 - x)^2 )^n; exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 53*x^2 + 554*x^3 + ... = Sum_{n >= 0} A066357(n+1)*x^n. - Peter Bala, Jun 23 2015

a(n) = Sum_{i = 0..n} binomial(4*n,i)*binomial(3*n-i-1,n-i). - Peter Bala, Sep 29 2015

a(n) = A000984(n)*Prod_{j=0, n} (2^j/(j!*(2*j-1)!!))*A068424(n, j)^2, with A068424 the falling factorial. See (5.4) in Podestá link. - Michel Marcus, Mar 31 2016

a(n) = GegenbauerC(2*n, -2*n, -1). - Peter Luschny, May 07 2016

a(n) = [x^n] 1/sqrt(1 - 4*x)^(2*n+1). - Ilya Gutkovskiy, Oct 10 2017

EXAMPLE

a(n) = (1/Pi)*int(x^(2n)/sqrt(4-(x-2)^2),x,0,4). - Paul Barry, Sep 17 2010

G.f. = 1 + 6*x + 70*x^2 + 924*x^3 + 12870*x^4 + 184756*x^5 + 2704156*x^6 + ...

MAPLE

A001448 := n-> binomial(4*n, 2*n) ;

MATHEMATICA

Table[Binomial[4n, 2n], {n, 0, 20}] (* Harvey P. Dale, Apr 26 2014 *)

a[ n_] := If[ n < 0, 0, HypergeometricPFQ[ {-2 n, -2 n}, {1}, 1]]; (* Michael Somos, Oct 22 2014 *)

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, A066357, A000984 (k = 0), A091527 (k = 1), A262732 (k = 3), A211419 (k = 4), A262733 (k = 5), A211421 (k = 6).

Sequence in context: A286527 A104900 A186667 * A024489 A036361 A182563

Adjacent sequences:  A001445 A001446 A001447 * A001449 A001450 A001451

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified February 23 22:52 EST 2018. Contains 299595 sequences. (Running on oeis4.)