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 A108475 Expansion of (1-3*x) / (1-5*x-5*x^2+x^3). 4
 1, 2, 15, 84, 493, 2870, 16731, 97512, 568345, 3312554, 19306983, 112529340, 655869061, 3822685022, 22280241075, 129858761424, 756872327473, 4411375203410, 25711378892991, 149856898154532, 873430010034205, 5090723162050694, 29670908962269963 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums of A108477. In general, Sum_{k=0..n} Sum_{j=0..n} binomial(2(n-k), j)*binomial(2k, j)*r^j has expansion (1-(r+1)*x)/((1 + (r+3)*x + (r-1)*(r+3)*x^2 + (r-1)^3*x^3). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,5,-1). FORMULA G.f.: (1-3*x)/((1+x)*(1-6*x+x^2)). a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(2(n-k), j)*binomial(2*k, j)2^j. Conjecture: a(n) = A000129(n+1)*A001333(n). - R. J. Mathar, Jul 08 2009 a(n) + a(n+1) = A001541(n+1). - R. J. Mathar, Jul 13 2009 a(n) = (4*(-1)^n - (3-2*sqrt(2))^n*(-2+sqrt(2)) + (2+sqrt(2))*(3+2*sqrt(2))^n)/8. - Colin Barker, Nov 04 2016 a(n) = (-1)^n * Re(sqrt(1+i) * cos((n + 1/2) * arccos(i)) * sin(n * arccos(i)) + 1), where i = sqrt(-1). - Daniel Suteu, Jun 23 2018 MATHEMATICA CoefficientList[Series[(1-3x)/(1-5x-5x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{5, 5, -1}, {1, 2, 15}, 30] (* Harvey P. Dale, Dec 30 2019 *) PROG (PARI) Vec((1-3*x)/((1+x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 04 2016 CROSSREFS Sequence in context: A057152 A002740 A178750 * A328007 A098624 A116079 Adjacent sequences:  A108472 A108473 A108474 * A108476 A108477 A108478 KEYWORD easy,nonn AUTHOR Paul Barry, Jun 04 2005 STATUS approved

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Last modified September 18 16:42 EDT 2020. Contains 337170 sequences. (Running on oeis4.)