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A290994 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^7.

%I A290994 #13 Apr 12 2023 08:06:43
%S A290994 0,0,0,0,0,0,1,7,28,84,210,462,924,1717,3017,5110,8568,14756,27132,
%T A290994 54264,116281,257775,572264,1246784,2641366,5430530,10861060,21242341,
%U A290994 40927033,78354346,150402700,291693136,574274008,1148548016,2326683921,4749439975
%N A290994 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^7.
%C A290994 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 A290994 See A291000 for a guide to related sequences.
%H A290994 Clark Kimberling, <a href="/A290994/b290994.txt">Table of n, a(n) for n = 0..1000</a>
%H A290994 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,2).
%F A290994 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + 2*a(n-7) for n >= 8.
%F A290994 G.f.: x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)). - _Colin Barker_, Aug 22 2017
%F A290994 a(n) = A049017(n-6) for n > 5. - _Georg Fischer_, Oct 23 2018
%F A290994 G.f.: x^6/((1-x)^7 - x^7). - _G. C. Greubel_, Apr 11 2023
%t A290994 z = 60; s = x/(1 - x); p = 1 - s^7;
%t A290994 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A290994 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290994 *)
%o A290994 (PARI) concat(vector(6), Vec(x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)) + O(x^50))) \\ _Colin Barker_, Aug 22 2017
%o A290994 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/((1-x)^7 - x^7) )); // _G. C. Greubel_, Apr 11 2023
%o A290994 (SageMath)
%o A290994 def A290994_list(prec):
%o A290994     P.<x> = PowerSeriesRing(ZZ, prec)
%o A290994     return P( x^6/((1-x)^7 - x^7) ).list()
%o A290994 A290994_list(60) # _G. C. Greubel_, Apr 11 2023
%Y A290994 Cf. A000012, A033453, A049017, A289780, A291000.
%Y A290994 Sequences of the form x^(m-1)/((1-x)^m - x^m): A000079 (m=1), A131577 (m=2), A024495 (m=3), A000749 (m=4), A139761 (m=5), A290993 (m=6), this sequence (m=7), A290995 (m=8).
%K A290994 nonn,easy
%O A290994 0,8
%A A290994 _Clark Kimberling_, Aug 22 2017