A232719 Sum_{k=1,...,2n} (-1)^k binomial(8*n,4*k).

-69, 9231, -1254465, 170459391, -23162405889, 3147359850495, -427670341173249, 58112808641953791, -7896499249846943745, 1072994093040913088511, -145800852665566628937729, 19811748057028406926114815, -2692064922113214275888611329, 365803841438484687010033303551

Offset

1,1

Comments

All elements of this sequence are multiples of 3. Any proof?

This follows from the recurrence relation. - Charles R Greathouse IV, Dec 13 2013

Links

Colin Barker, Table of n, a(n) for n = 1..450

Index entries for linear recurrences with constant coefficients, signature (-135,120,16).

Formula

a(n) = (-1)^n/2 * ((2+sqrt(2))^(4*n) + (2-sqrt(2))^(4*n)) - 1. - Vaclav Kotesovec, Dec 06 2013

G.f.: 3*x*(28*x+23) / ((x-1)*(16*x^2+136*x+1)). - Colin Barker, Dec 06 2013

Maple

A232719:=n->add((-1)^i*binomial(8*n, 4*i), i=1..2*n); seq(A232719(n), n=1..20); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

A[n_] := Sum[(-1)^k Binomial[8 n, 4 k], {k, 1, 2n}]; Array[A, 33]
Table[FullSimplify[(-1)^n/2*((2+Sqrt[2])^(4*n)+(2-Sqrt[2])^(4*n))-1], {n, 1, 15}] (* Vaclav Kotesovec, Dec 06 2013 *)
CoefficientList[Series[3 (28 x + 23) / ((x - 1) (16 x^2 + 136 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)
LinearRecurrence[{-135, 120, 16}, {-69, 9231, -1254465}, 20] (* Harvey P. Dale, Jan 31 2023 *)

Prog

(PARI) a(n)=sum(k=1, 2*n, (-1)^k*binomial(8*n, 4*k)) \\ Charles R Greathouse IV, Dec 13 2013
(PARI) Vec(3*x*(28*x+23)/((x-1)*(16*x^2+136*x+1)) + O(x^100)) \\ Colin Barker, Nov 09 2014

Crossrefs

Cf. A232732.

Sequence in context: A297550 A049000 A234824 * A297807 A251002 A194612

Adjacent sequences: A232716 A232717 A232718 * A232720 A232721 A232722

Keyword

sign,easy

Author

José María Grau Ribas, Nov 28 2013

Status

approved