A084151 Binomial transform of a Pell convolution.

0, 0, 1, 9, 62, 390, 2359, 14007, 82412, 482652, 2820061, 16457397, 95983370, 559619970, 3262267891, 19015581699, 110836005272, 646014798840, 3765295834489, 21945889348257, 127910427675542, 745517838966462

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A006668. Second binomial transform of A084150.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-19,3).

Crossrefs

Cf. A000244, A001541, A006668, A084150.

Programs

Magma

[(Evaluate(ChebyshevFirst(n), 2) -3^n)/8: n in [0..40]]; // G. C. Greubel, Oct 11 2022

Mathematica

LinearRecurrence[{9,-19,3},{0,0,1},30] (* Harvey P. Dale, Jun 06 2021 *)

SageMath

[(chebyshev_T(n, 3) - 3^n)/8 for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = ( (3+sqrt(8))^n + (3-sqrt(8))^n - 2*3^n )/16.

E.g.f.: (1/4)*exp(3*x)*( sinh(sqrt(2)*x) )^2.

G.f.: x^2 / ( (1-3*x)*(1-6*x+x^2) ). - R. J. Mathar, Sep 27 2012

a(n) = (A001541(n) - 3^n)/8. - R. J. Mathar, Sep 27 2012

a(n) = (1/8)*(ChebyshevT(n, 3) - 3^n) = (A001541(n) - A000244(n))/8. - G. C. Greubel, Oct 11 2022