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%I #11 Feb 24 2022 19:01:57
%S 1,3,10,32,102,323,1021,3224,10177,32121,101378,319960,1009830,
%T 3187145,10059029,31747584,100199485,316242607,998102878,3150142840,
%U 9942261690,31379074783,99036453193,312571964808,986517893269,3113579153493,9826861945870
%N p-INVERT of A010892, where p(S) = 1 - S - S^2 - S^3.
%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).
%H Clark Kimberling, <a href="/A292398/b292398.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4, -1, -6, 1, 4, 1)
%F G.f.: -((1 - x - x^2 + x^3 + x^4)/(-1 + 4 x - x^2 - 6 x^3 + x^4 + 4 x^5 + x^6)).
%F a(n) = 4*a(n-1) - a(n-2) - 6*a(n-3) + a(n-4) + 4*a(n-5) + a(n-6) for n >= 7.
%p A292398:=proc(n) option remember:
%p if n=0 then 1 elif n=1 then 3 elif n=2 then 10 elif n=3 then 32 elif n=4 then 102 elif n=5 then 323 elif n>=6 then 4*procname(n-1)-procname(n-2)-6*procname(n-3)+procname(n-4)+4*procname(n-5)+procname(n-6) fi; end:
%p seq(A292398(n),n=0..10^2); # _Muniru A Asiru_, Oct 02 2017
%t z = 60; s = x/(1 - x + x^2); p = 1 - s - s^2 - s^3;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A010892 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292398 *)
%t LinearRecurrence[{4,-1,-6,1,4,1},{1,3,10,32,102,323},30] (* _Harvey P. Dale_, Feb 24 2022 *)
%o (GAP)
%o a:=[1,3,10,32,102,323];; for n in [7..10^2] do a[n]:=4*a[n-1]-a[n-2]-6*a[n-3]+a[n-4]+4*a[n-5]+a[n-6]; od; A292398:=a; # _Muniru A Asiru_, Oct 02 2017
%o (PARI) Vec(-(1 - x - x^2 + x^3 + x^4)/(-1 + 4*x - x^2 - 6*x^3 + x^4 + 4*x^5 + x^6) + O(x^20)) \\ _Felix Fröhlich_, Oct 02 2017
%Y Cf. A010892, A292301.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Sep 29 2017
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