A084901 a(n) = 4^(n-2)*n*(5*n+3)/2.

0, 1, 13, 108, 736, 4480, 25344, 136192, 704512, 3538944, 17367040, 83623936, 396361728, 1853882368, 8573157376, 39258685440, 178241142784, 803158884352, 3594887626752, 15994458210304, 70781061038080, 311711546474496

Offset: 0

Author

Paul Barry, Jun 10 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A084900. Third binomial transform of heptagonal numbers A000566. Fourth binomial transform of (0,1,5,0,0,0,...).

Coefficients in the hypergeometric series identity 1 - 13*x/(x + 12) + 108*x*(x - 1)/((x + 12)*(x + 16)) - 736*x*(x - 1)*(x - 2)/((x + 12)*(x + 16)*(x + 20)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A077616. - Peter Bala, May 30 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Crossrefs

Cf. A084902, A077616, A276289.

Programs

GAP

List([0..30], n-> 2^(2*n-5)*n*(5*n+3)); # G. C. Greubel, Jun 06 2019

Magma

[2^(2*n-5)*n*(5*n+3): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Mathematica

Table[2^(2*n-5)*n*(5*n+3), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{12,-48,64},{0,1,13},30] (* or *) CoefficientList[ Series[-((x (1+x))/(-1+4 x)^3),{x,0,30}],x] (* Harvey P. Dale, Jul 14 2021 *)

PARI

vector(30, n, n--; 2^(2*n-5)*n*(5*n+3)) \\ G. C. Greubel, Jun 06 2019

Sage

[2^(2*n-5)*n*(5*n+3) for n in (0..30)] # G. C. Greubel, Jun 06 2019

Formula

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

E.g.f.: x*(2 + 5*x)*exp(4*x)/2. - G. C. Greubel, Jun 06 2019

a(n) = 12*a(n-1)-48*a(n-2)+64*a(n-3). - Wesley Ivan Hurt, May 28 2021