This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A290995 #12 Apr 12 2023 08:06:39
%S A290995 0,0,0,0,0,0,0,1,8,36,120,330,792,1716,3432,6436,11456,19584,32640,
%T A290995 54264,93024,170544,341088,735472,1653632,3749760,8386560,18289440,
%U A290995 38724480,79594560,159189120,311058496,597137408,1133991936,2147450880,4089171840
%N A290995 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^8.
%C A290995 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 A290995 See A291000 for a guide to related sequences.
%H A290995 Clark Kimberling, <a href="/A290995/b290995.txt">Table of n, a(n) for n = 0..1000</a>
%H A290995 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8).
%F A290995 a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) for n >= 9.
%F A290995 G.f.: x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)). - _Colin Barker_, Aug 22 2017
%F A290995 G.f.: x^7/((1-x)^8 - x^8). - _G. C. Greubel_, Apr 11 2023
%t A290995 z = 60; s = x/(1 - x); p = 1 - s^8;
%t A290995 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)
%t A290995 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290995 *)
%o A290995 (PARI) concat(vector(7), Vec(x^7 / ((1 - 2*x)*(1 - 2*x + 2*x^2)*(1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4)) + O(x^50))) \\ _Colin Barker_, Aug 22 2017
%o A290995 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0,0] cat Coefficients(R!( x^7/((1-x)^8 - x^8) )); // _G. C. Greubel_, Apr 11 2023
%o A290995 (SageMath)
%o A290995 def A290995_list(prec):
%o A290995 P.<x> = PowerSeriesRing(ZZ, prec)
%o A290995 return P( x^7/((1-x)^8 - x^8) ).list()
%o A290995 A290995_list(60) # _G. C. Greubel_, Apr 11 2023
%Y A290995 Cf. A000012, A033453, A289780, A291000.
%Y A290995 Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), this sequence (m=8), A306939 (m=9).
%K A290995 nonn,easy
%O A290995 0,9
%A A290995 _Clark Kimberling_, Aug 22 2017