cp's OEIS Frontend

A172485 The case S(-1,-2,3) of the family of self-convolutive recurrences studied by Martin and Kearney.

%I A172485 #26 Jul 03 2023 11:23:56
%S A172485 1,-1,-1,3,3,-21,27,27,-117,27,459,-837,-405,3483,-2997,-9477,25515,
%T A172485 -1701,-91125,130491,165483,-732645,422091,2136699,-4546773,-1712421,
%U A172485 19597707,-21237957,-43597845,140635035,-32890293,-483552261,804188331
%N A172485 The case S(-1,-2,3) of the family of self-convolutive recurrences studied by Martin and Kearney.
%H A172485 Vincenzo Librandi, <a href="/A172485/b172485.txt">Table of n, a(n) for n = 1..1000</a>
%H A172485 R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%H A172485 R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 311.
%H A172485 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3, -6, -6).
%F A172485 a(n) = - (n + 2) * a(n-1) + 3 * Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - _Michael Somos_, Jul 24 2011
%F A172485 G.f.: (x + 2*x^2 + 2*x^3) / (1 + 3*x + 6*x^2 + 6*x^3).
%F A172485 a(n) = -3 * (a(n-1) + 2*a(n-2) + 2*a(n-3)) if n>3.
%F A172485 G.f.: x / (1 + x / (1 - 2*x / (1 + 2*x / (1 - x / (1 + 3*x))))). - _Michael Somos_, Jan 03 2013
%e A172485 G.f. = x - x^2 - x^3 + 3*x^4 + 3*x^5 - 21*x^6 + 27*x^7 + 27*x^8 - 117*x^9 + ...
%t A172485 CoefficientList[Series[1/(1 + x/(1 - 2*x/(1 + 2*x/(1 - x/(1 + 3*x))))), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 03 2013 *)
%o A172485 (PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = -(k + 2) * A[k-1] + 3 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 24 2011 */
%o A172485 (Magma) I:=[1,-1,-1,3,3,-21]; [n le 6 select I[n] else -3*(Self(n-1)+2*Self(n-2)+2*Self(n-3)): n in [1..40]]; // _Vincenzo Librandi_, Jan 03 2013
%K A172485 sign,easy
%O A172485 1,4
%A A172485 _N. J. A. Sloane_, Nov 20 2010