A055787 a(n) = 2^(4*n-1) - 2^(2*n-1)*binomial(2*n,n).

0, 4, 80, 1408, 23808, 395264, 6496256, 106102784, 1725759488, 27987017728, 452890460160, 7316694892544, 118053383700480, 1902813588226048, 30644438111879168, 493182667873845248, 7932561728196313088, 127528427796392050688, 2049363966872621416448

Offset

0,2

Comments

Number of 4-ary words of length 2n in which the combined count of 0's and 1's is greater than n. - David Scambler, Aug 13 2012

References

The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1982, (3.98), page 33.

Links

G. C. Greubel, Table of n, a(n) for n = 0..825

H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010, (7.6) p. 35.

Formula

a(n) + A045952(n) = 2^(4n). - David Scambler, Aug 13 2012

a(n) = Sum_{k=0..n-1} binomial(4*k+2, 2*k+1) * binomial(4*n-4*k-2, 2*n-2*k-1). (LHS of Gould's identity, corrected).

a(n) = 2^(4*n-1)*(1-(Gamma(n+1/2))/(sqrt(Pi)*Gamma(n+1))). - Alexander R. Povolotsky, Aug 13 2012

a(n) = Sum_{k=0..n-1}(2^(2*n) * C(2*n, k)). - David Scambler, Aug 13 2012

a(n+2) = (8*(4*n+5)*a(n+1))/(n+2) - (128*(2*n+1)*a(n))/(n+2). - Alexander R. Povolotsky, Aug 14 2012

G.f.: (1 - sqrt(1-16*x))/(2*(1-16*x)). - G. C. Greubel, Jan 25 2020

Maple

seq( 2^(4*n-1) - 2^(2*n-1)*binomial(2*n, n), n=0..20); # G. C. Greubel, Jan 25 2020

Mathematica

Table[2^(4*n-1) - 2^(2*n-1)*Binomial[2*n, n], {n, 0, 20}] (* G. C. Greubel, Feb 16 2017 *)

Prog

(PARI) vector(21, n, my(m=n-1); 2^(4*m-1) -2^(2*m-1)*binomial(2*m, m) ) \\ G. C. Greubel, Feb 16 2017
(Magma) [2^(4*n-1) - 2^(2*n-1)*Binomial(2*n, n): n in [0..20]]; // G. C. Greubel, Jan 25 2020
(Sage) [2^(4*n-1) - 2^(2*n-1)*binomial(2*n, n) for n in (0..20)] # G. C. Greubel, Jan 25 2020
(GAP) List([0..20], n-> 2^(4*n-1) - 2^(2*n-1)*Binomial(2*n, n)); # G. C. Greubel, Jan 25 2020

Crossrefs

Cf. A045952.

Sequence in context: A192834 A054322 A114488 * A132584 A277074 A012127

Adjacent sequences: A055784 A055785 A055786 * A055788 A055789 A055790

Keyword

nonn

Author

N. J. A. Sloane, Jul 15 2000

Status

approved