A133474 -id:A133474 - cp's OEIS frontend

A139797 Inverse binomial transform of [0, A133474].

0, 0, 0, 0, 1, 1, 3, 4, 10, 18, 39, 75, 153, 302, 608, 1212, 2429, 4853, 9711, 19416, 38838, 77670, 155347, 310687, 621381, 1242754, 2485516, 4971024, 9942057, 19884105, 39768219, 79536428, 159072866, 318145722, 636291455, 1272582899, 2545165809, 5090331606, 10180663224, 20361326436, 40722652885, 81445305757

Offset: 0

Paul Curtz, May 22 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A010892.

f:= func< n | Evaluate(ChebyshevU(n+1), 1/2) >;
[n eq 0 select 0 else ((3*n-4)*(-1)^n +2^n +3*f(n) -6*f(n-1))/27: n in [0..60]]; // G. C. Greubel, Mar 08 2021

Table[((3*n-4)*(-1)^n +2^n +3*ChebyshevU[n, 1/2] -6*ChebyshevU[n-1, 1/2])/27, {n, 0, 60}] (* G. C. Greubel, Mar 08 2021 *)

[( (3*n-4)*(-1)^n +2^n +3*chebyshev_U(n, 1/2) -6*chebyshev_U(n-1, 1/2) )/27 for n in (0..60)] # G. C. Greubel, Mar 08 2021

G.f.: x^4/((1+x)^2 * (1-2*x) * (1-x+x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = ( (3*n-4)*(-1)^n +2^n +3*ChebyshevU(n, 1/2) -6*ChebyshevU(n-1, 1/2) )/27. - G. C. Greubel, Mar 08 2021

Edited by R. J. Mathar, Sep 08 2009

Terms a(29) onward added by G. C. Greubel, Mar 08 2021

A135254 Binomial transform of A131666.

Original entry on oeis.org

0, 0, 1, 4, 12, 33, 90, 252, 729, 2160, 6480, 19521, 58806, 176904, 531441, 1595052, 4785156, 14353281, 43053282, 129146724, 387420489, 1162241784, 3486725352, 10460235105, 31380882462, 94143001680, 282429536481, 847289140884

Offset: 0

Paul Curtz, Nov 30 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,9).

Cf. A133474.

a:=[0,1,4];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; Concatenation([0], a); # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 21 2019

CoefficientList[Series[x^2(2x-1)/((3x^2-3x+1)(3x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-12,9},{0,0,1,4},30] (* Harvey P. Dale, May 26 2011 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x)))) \\ G. C. Greubel, Nov 21 2019

def A135254_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^2*(1-2*x)/((1-3*x+3*x^2)*(1-3*x))).list()
A135254_list(30) # G. C. Greubel, Nov 21 2019

From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: x^2*(1-2*x)/((1 - 3*x + 3*x^2)*(1-3*x)).
a(n) = 6*a(n-1) - 12*a(n-2) + 9*a(n-3). (End)

More terms from R. J. Mathar, Apr 02 2008

cp's OEIS Frontend

A139797 Inverse binomial transform of [0, A133474].

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions