Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Sep 03 2017 21:41:36
%S 1,2,5,12,27,65,146,346,788,1845,4239,9865,22758,52818,122072,282954,
%T 654528,1516221,3508817,8125763,18808494,43550500,100815652,233418699,
%U 540371471,1251079052,2896357943,6705591388,15524220275,35941069252,83208225215
%N p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S - S^2 - S^3 - S^4 + S^5.
%C Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
%C See A291219 for a guide to related sequences.
%H Clark Kimberling, <a href="/A291248/b291248.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1, 6, -3, -12, 3, 12, -3, -6, 1, 1)
%F G.f.: -((1 + x - 3 x^2 - 2 x^3 + 3 x^4 + 2 x^5 - 3 x^6 - x^7 + x^8)/((-1 - x + x^2) (1 - 2 x - 3 x^2 + 4 x^3 + 5 x^4 - 4 x^5 - 3 x^6 + 2 x^7 + x^8))).
%F a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 12*a(n-4) + 3*a(n-5) + 12*a(n-6) - 3*a(n-7) - 6*a(n-8) + a(n-9) + a(n-10) for n >= 11.
%t z = 60; s = x/(1 - x^2); p = 1 - s - s^2 - s^3 - s^4 + s^5;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291248 *)
%Y Cf. A000035, A291219.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Aug 29 2017