A080930 a(n) = 2^(n-3)*(n+2)*(n+3)*(n+4)/3.

1, 5, 20, 70, 224, 672, 1920, 5280, 14080, 36608, 93184, 232960, 573440, 1392640, 3342336, 7938048, 18677760, 43581440, 100925440, 232128512, 530579456, 1205862400, 2726297600, 6134169600, 13740539904, 30651973632, 68115496960

Offset: 0

Author

Paul Barry, Feb 26 2003

Keywords

• nonn
• easy

Comments

Old definition was "Sequence associated with recurrence a(n)=2*a(n-1)+k(k+2)*a(n-2)". See the first comment in A080928.

The fourth column of triangle A080928 (after 0) is 4*a(n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Crossrefs

Cf. A080928.

Programs

GAP

List([0..30], n-> 2^(n-2)*Binomial(n+4,3)); # G. C. Greubel, Aug 27 2019

Magma

[Binomial(n+3,3)*2^(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Maple

[seq (binomial(n+3,3)*2^(n-3),n=1..27)]; # Zerinvary Lajos, Oct 29 2006

Mathematica

CoefficientList[Series[(1-x)(1 -2x +2x^2)/(1-2x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8, -24, 32, -16}, {1, 5, 20, 70}, 30] (* Bruno Berselli, Aug 06 2013 *)

PARI

a(n)=2^(n-3)*(n+2)*(n+3)*(n+4)/3 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[2^(n-2)*binomial(n+4,3) for n in (0..30)] # G. C. Greubel, Aug 27 2019

Formula

G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^4 = (1-3*x+4*x^2-2*x^3)/(1-2*x)^4.

a(n) = binomial(n+3,3)*2^(n-3), n>0. - Zerinvary Lajos, Oct 29 2006

a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n>3, a(0)=1, a(1)=5, a(2)=20, a(3)=70. - Bruno Berselli, Aug 06 2013

E.g.f.: (3 +9*x +6*x^2 +x^3)*exp(2*x)/3. - G. C. Greubel, Aug 27 2019

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 48*log(2) - 32.

Sum_{n>=0} (-1)^n/a(n) = 176 - 432*log(3/2). (End)

Extensions

Edited by Bruno Berselli, Aug 06 2013