A100098 -id:A100098 - cp's OEIS frontend

A127361 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).

1, -1, 4, -7, 22, -46, 130, -295, 790, -1870, 4864, -11782, 30148, -73984, 187534, -463687, 1168870, -2902870, 7293640, -18161170, 45541492, -113576596, 284470564, -710118262, 1777323772, -4439253196, 11105933440, -27749232700, 69403169200

Offset: 0

Paul Barry, Jan 11 2007

Hankel transform is 3^n. In general, for r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+2*x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4*x^2)) * g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
Second binomial transform is A026641. - Philippe Deléham, Mar 14 2007
Signed version of A100098. - Philippe Deléham, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A026641, A061554.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1+2*x-Sqrt(1-4*x^2))/(2*Sqrt(1-4*x^2)*(1+x-Sqrt(1-4*x^2))) )); // G. C. Greubel, Feb 17 2019

a:=n->add(binomial(n,floor(k/2))*(-2)^(n-k),k=0..n): seq(a(n),n=0..30); # Muniru A Asiru, Feb 18 2019

CoefficientList[Series[(1/Sqrt[1-4*x^2])*(1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) /(1+2*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

my(x='x+O('x^30)); Vec( (1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ) \\ G. C. Greubel, Feb 17 2019

((1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019

G.f.: (1/sqrt(1-4*x^2))(1+x*c(x^2))/(1+2*x*c(x^2)), with c(x) = (1 - sqrt(1-4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Nov 25 2007
a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - Philippe Deléham, Dec 04 2009
Conjecture: 2*n*a(n) + (5*n-4)*a(n-1) - 2*(4*n-3)*a(n-2) - 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ (-1)^n * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014

More terms from Vincenzo Librandi, Feb 15 2014

A112657 A Motzkin transform of Jacobsthal numbers.

Original entry on oeis.org

1, 2, 7, 23, 79, 272, 943, 3278, 11419, 39830, 139057, 485795, 1697905, 5936348, 20760271, 72615143, 254028355, 888758030, 3109714117, 10881403229, 38077702909, 133251869648, 466325356273, 1631981113112, 5711490384901

Offset: 0

Paul Barry, Jan 11 2006

Binomial transform of A100098.

Inverse binomial transform of A007854. The Hankel transform of this sequence is 3^n (see A000244). - Philippe Deléham, Nov 25 2007

Cf. A000244, A007854, A026300, A089942, A100098.

a(n) = Sum_{k=0..n} A026300(n, k)*(2^(k+1) + (-1)^k)/3, where A026300 is the Motzkin triangle; a(n) = Sum_{k=0..n} ((k+1)/(n+1))*Sum_{j=0..n+1} C(n+1, j)*C(j, 2j-n+k)*(2^(k+1) + (-1)^k)/3.

a(n) = Sum_{k=0..n} A089942(n,k)*2^k = Sum_{k=0..n} A071947(n,k)*2^(n-k). - Philippe Deléham, Mar 31 2007

cp's OEIS Frontend

A127361 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions