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 A290890 #17 May 29 2022 18:07:05
%S A290890 0,1,4,11,28,72,188,493,1292,3383,8856,23184,60696,158905,416020,
%T A290890 1089155,2851444,7465176,19544084,51167077,133957148,350704367,
%U A290890 918155952,2403763488,6293134512,16475640049,43133785636,112925716859,295643364940,774004377960
%N A290890 p-INVERT of the positive integers, where p(S) = 1 - S^2.
%C A290890 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 A290890 Note that in A290890, s = (1,2,3,4,...); i.e., A000027(n+1) for n>=0, whereas in A290990, s = (0,1,2,3,4,...); i.e., A000027(n) for n>=0.
%C A290890 Guide to p-INVERT sequences using s = (1,2,3,4,5,...) = A000027:
%C A290890 p(S) t(1,2,3,4,5,...)
%C A290890 1 - S A001906
%C A290890 1 - S^2 A290890; see A113067 for signed version
%C A290890 1 - S^3 A290891
%C A290890 1 - S^4 A290892
%C A290890 1 - S^5 A290893
%C A290890 1 - S^6 A290894
%C A290890 1 - S^7 A290895
%C A290890 1 - S^8 A290896
%C A290890 1 - S - S^2 A289780
%C A290890 1 - S - S^3 A290897
%C A290890 1 - S - S^4 A290898
%C A290890 1 - S^2 - S^4 A290899
%C A290890 1 - S^2 - S^3 A290900
%C A290890 1 - S^3 - S^4 A290901
%C A290890 1 - 2S A052530; (1/2)*A052530 = A001353
%C A290890 1 - 3S A290902; (1/3)*A290902 = A004254
%C A290890 1 - 4S A003319; (1/4)*A003319 = A001109
%C A290890 1 - 5S A290903; (1/5)*A290903 = A004187
%C A290890 1 - 2*S^2 A290904; (1/2)*A290904 = A290905
%C A290890 1 - 3*S^2 A290906; (1/3)*A290906 = A290907
%C A290890 1 - 4*S^2 A290908; (1/4)*A290908 = A099486
%C A290890 1 - 5*S^2 A290909; (1/5)*A290909 = A290910
%C A290890 1 - 6*S^2 A290911; (1/6)*A290911 = A290912
%C A290890 1 - 7*S^2 A290913; (1/7)*A290913 = A290914
%C A290890 1 - 8*S^2 A290915; (1/8)*A290915 = A290916
%C A290890 (1 - S)^2 A290917
%C A290890 (1 - S)^3 A290918
%C A290890 (1 - S)^4 A290919
%C A290890 (1 - S)^5 A290920
%C A290890 (1 - S)^6 A290921
%C A290890 1 - S - 2*S^2 A290922
%C A290890 1 - 2*S - 2*S^2 A290923; (1/2)*A290923 = A290924
%C A290890 1 - 3*S - 2*S^2 A290925
%C A290890 (1 - S^2)^2 A290926
%C A290890 (1 - S^2)^3 A290927
%C A290890 (1 - S^3)^2 A290928
%C A290890 (1 - S)(1 - S^2) A290929
%C A290890 (1 - S^2)(1 - S^4) A290930
%C A290890 1 - 3 S + S^2 A291025
%C A290890 1 - 4 S + S^2 A291026
%C A290890 1 - 5 S + S^2 A291027
%C A290890 1 - 6 S + S^2 A291028
%C A290890 1 - S - S^2 - S^3 A291029
%C A290890 1 - S - S^2 - S^3 - S^4 A201030
%C A290890 1 - 3 S + 2 S^3 A291031
%C A290890 1 - S - S^2 - S^3 + S^4 A291032
%C A290890 1 - 6 S A291033
%C A290890 1 - 7 S A291034
%C A290890 1 - 8 S A291181
%C A290890 1 - 3 S + 2 S^3 A291031
%C A290890 1 - 3 S + 2 S^2 A291182
%C A290890 1 - 4 S + 2 S^3 A291183
%C A290890 1 - 4 S + 3 S^3 A291184
%H A290890 Clark Kimberling, <a href="/A290890/b290890.txt">Table of n, a(n) for n = 0..1000</a>
%H A290890 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -5, 4, -1)
%F A290890 G.f.: x/(1 - 4 x + 5 x^2 - 4 x^3 + x^4).
%F A290890 a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - a(n-4).
%e A290890 (See the examples at A289780.)
%t A290890 z = 60; s = x/(1 - x)^2; p = 1 - s^2;
%t A290890 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t A290890 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290890 *)
%Y A290890 Cf. A000027, A113067, A289780, A113067 (signed version of same sequence).
%K A290890 nonn,easy
%O A290890 0,3
%A A290890 _Clark Kimberling_, Aug 15 2017