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A232732 Sum_{k=0,...,2n} (-1)^k binomial(12*n,6*k). 2
-922, 2434966, -6575675482, 17767242206806, -48007067114436442, 129715076631793674646, -350490088991612425827802, 947024090736392816800774486, -2558858742679396890519761433562, 6914035375695623821224314247122326, -18681721026270831871754901657845477722 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(2n) == 1 (mod 3); a(2n+1) == 2 (mod 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..290

Index entries for linear recurrences with constant coefficients, signature (-2766,-172929,-64).

FORMULA

a(n) = (-1)^n/3 * (2^(6*n) + ((sqrt(3)+1)^(12*n) + (sqrt(3)-1)^(12*n))/2^(6*n)). - Vaclav Kotesovec, Dec 06 2013

G.f.: -2*x*(32*x^2+57643*x+461) / ((64*x+1)*(x^2+2702*x+1)). - Colin Barker, Dec 06 2013

MAPLE

A232732:=n->add((-1)^i*binomial(12*n, 6*i), i=0..2*n); seq(A232732(n), n=1..20); # Wesley Ivan Hurt, Dec 06 2013

MATHEMATICA

A[n_] := Sum[(-1)^k Binomial[12 n, 6 k], {k, 0, 2n}]; Array[A, 14]

CoefficientList[Series[-2 (32 x^2 + 57643 x + 461) / ((64 x + 1) (x^2 + 2702 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 09 2014 *)

LinearRecurrence[{-2766, -172929, -64}, {-922, 2434966, -6575675482}, 20] (* Harvey P. Dale, Apr 18 2019 *)

PROG

(PARI) a(n)=sum(k=0, 2*n, (-1)^k*binomial(12*n, 6*k)) \\ Charles R Greathouse IV, Dec 13 2013

(PARI) Vec(-2*x*(32*x^2+57643*x+461)/((64*x+1)*(x^2+2702*x+1)) + O(x^100)) \\ Colin Barker, Nov 09 2014

CROSSREFS

Cf. A232719.

Sequence in context: A289794 A235881 A051984 * A228673 A268849 A177810

Adjacent sequences:  A232729 A232730 A232731 * A232733 A232734 A232735

KEYWORD

sign,easy

AUTHOR

José María Grau Ribas, Nov 29 2013

STATUS

approved

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Last modified June 1 01:37 EDT 2020. Contains 334757 sequences. (Running on oeis4.)