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%I #6 Aug 28 2017 09:19:05
%S 10,65,360,1831,8830,41104,186710,833401,3672840,16034303,69506930,
%T 299700192,1287010850,5509712833,23531008200,100312445063,
%U 427025152550,1815832379312,7714875191470,32756357939033,139008007848360,589672772732671,2500620567692890
%N p-INVERT of (0,1,0,1,0,1,...), where p(S) = (1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S).
%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="/A291231/b291231.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (10, -31, 20, 40, -20, -31, -10, -1)
%F G.f.: (10 - 35 x + 20 x^2 + 46 x^3 - 20 x^4 - 35 x^5 - 10 x^6)/(1 - 10 x + 31 x^2 - 20 x^3 - 40 x^4 + 20 x^5 + 31 x^6 + 10 x^7 + x^8).
%F a(n) = 10*a(n-1) - 31*a(n-2) + 20*a(n-3) + 40*a(n-4) - 20*a(n-5) -31*a(n-6) - 10*a(n-7) - a(n-8) for n >= 7.
%t z = 60; s = x/(1 - x^2); p = (1 - s)(1 - 2 s)(1 - 3 s)(1 - 4 s);
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291231 *)
%Y Cf. A000035, A054447, A291219.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Aug 26 2017
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