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A290896 p-INVERT of the positive integers, where p(S) = 1 - S^8.

%I A290896 #10 Aug 17 2017 15:59:57
%S A290896 0,0,0,0,0,0,0,1,16,136,816,3876,15504,54264,170544,490315,1307536,
%T A290896 3269288,7732144,17436220,37819152,79883544,167737776,362063944,
%U A290896 839161648,2158258904,6136548496,18586871324,57486027952,176258492200,527387147664,1529591016109
%N A290896 p-INVERT of the positive integers, where p(S) = 1 - S^8.
%C A290896 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 A290896 See A290890 for a guide to related sequences.
%H A290896 Clark Kimberling, <a href="/A290896/b290896.txt">Table of n, a(n) for n = 0..1000</a>
%H A290896 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12869, 11440, -8008, 4368, -1820, 560, -120, 16, -1)
%F A290896 a(n) = 16*a(n-1) - 120*a(n-2) + 560*a(n-3) - 1820*a(n-4) + 4368*a(n-5) - 8008*a(n-6) + 11440*a(n-7) - 12869*a(n-8) + 11440*a(n-9) - 8008*a(n-10) + 4368*a(n-11) - 1820*a(n-12) + 560*a(n-13) - 120*a(n-14) + 16*a(n-15) - a(n-16).
%F A290896 G.f.: x^7 / ((1 - 3*x + x^2)*(1 - x + x^2)*(1 - 4*x + 7*x^2 - 4*x^3 + x^4)*(1 - 8*x + 28*x^2 - 56*x^3 + 71*x^4 - 56*x^5 + 28*x^6 - 8*x^7 + x^8)). - _Colin Barker_, Aug 16 2017
%t A290896 z = 60; s = x/(1 - x)^2; p = 1 - s^8;
%t A290896 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t A290896 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290896 *)
%o A290896 (PARI) concat(vector(7), Vec(x^7 / ((1 - 3*x + x^2)*(1 - x + x^2)*(1 - 4*x + 7*x^2 - 4*x^3 + x^4)*(1 - 8*x + 28*x^2 - 56*x^3 + 71*x^4 - 56*x^5 + 28*x^6 - 8*x^7 + x^8)) + O(x^40))) \\ _Colin Barker_, Aug 16 2017
%Y A290896 Cf. A000027, A290890.
%K A290896 nonn,easy
%O A290896 0,9
%A A290896 _Clark Kimberling_, Aug 15 2017