A073382 Fifth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

1, 12, 90, 532, 2709, 12432, 52808, 211248, 805374, 2951576, 10465476, 36079848, 121412942, 400054032, 1293997416, 4117416496, 12910964103, 39956039172, 122193599822, 369685154076, 1107503284923, 3288114790112

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (12,-54,100,-15,-168,76,168,-15,-100,-54,-12,-1).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073381(k).

a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+5, 5) * binomial(n-k, k).

a(n) = ((50085 +53006*n +19594*n^2 +3016*n^3 +164*n^4)*(n+1)*U(n+1) +(11355 +16336*n +7042*n^2 +1184*n^3 +68*n^4)*(n+2)*U(n))/(3*5*2^13), with U(n) = A000129(n+1), n >= 0.

G.f.: 1/(1-(2+x)*x)^6.

a(n) = F'''''(n+6, 2)/5!, that is, 1/5! times the 5th derivative of the (n+6)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Example

x^3 + 12*x^4 + 90*x^5 + 532*x^6 + 2709*x^7 + 12432*x^8 + 52808*x^9 + 211248*x^10 + 805374*x^11 +......+ 122193599822*x^21 + 369685154076*x^22 + 1107503284923*x^23 + 3288114790112*x^24 + etc. [Zerinvary Lajos, Jun 01 2009]

Mathematica

CoefficientList[Series[1/(1-2*x-x^2)^6, {x, 0, 40}], x] (* G. C. Greubel, Oct 02 2022 *)

Prog

(Sage) taylor( 1/(1-2*x-x^2)^6, x, 0, 24).list() # G. C. Greubel, Oct 02 2022
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^6 )); // G. C. Greubel, Oct 02 2022

Crossrefs

Sixth (m=5) column of triangle A054456, A073381.

Cf. A000129.

Sequence in context: A084485 A130072 A135158 * A036216 A022640 A090749

Adjacent sequences: A073379 A073380 A073381 * A073383 A073384 A073385

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Status

approved