A114014 Expansion of g.f. (1 + 2*x)^4/((1 + x)*(1 - 16*x^2)).

1, 7, 33, 127, 529, 2031, 8465, 32495, 135441, 519919, 2167057, 8318703, 34672913, 133099247, 554766609, 2129587951, 8876265745, 34073407215, 142020251921, 545174515439, 2272324030737, 8722792247023, 36357184491793

Offset: 0

Author

Roger L. Bagula, Jan 31 2006

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,16,16).

Crossrefs

Cf. A112627.

Programs

Magma

[1,7] cat [(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n): n in [2..40]]; // G. C. Greubel, Jul 07 2021

Mathematica

CoefficientList[Series[(1+2*x)^4/((1+x)*(1-16*x^2)), {x, 0, 40}], x]
a[n_]:= a[n]= If[n<2, 7^n, If[n==2, 33, 4*a[n-1] +(-1)^n*(4^(n-1) -1)/3]];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 07 2021 *)
LinearRecurrence[{-1,16,16},{1,7,33,127,529},30] (* Harvey P. Dale, Aug 07 2023 *)

Sage

[1,7]+[(1/30)*(4^(n-1)*(243 + 5*(-1)^n) - 2*(-1)^n) for n in (2..40)] # G. C. Greubel, Jul 07 2021

Formula

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

From Colin Barker, Dec 03 2012: (Start)

a(n) = (5*(-4)^n - 8*(-1)^n + 243*4^n)/120 for n>1.

G.f.: (1 +8*x +24*x^2 +32*x^3 +16*x^4)/((1+x)*(1-4*x)*(1+4*x)). (End)

From G. C. Greubel, Jul 07 2021: (Start)

a(n) = 4*a(n-1) + (-1)^n*(4^(n-1) -1)/3, n>2, with a(0) = 1, a(1) = 7, and a(2) = 33.

E.g.f.: (1/120)(243*exp(4*x) + 5*exp(-4*x) - 8*exp(-x) - 120*(1 + x)). (End)

Extensions

New name and edited by G. C. Greubel, Jul 07 2021