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p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^6.
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%I #27 Apr 12 2023 08:06:27

%S 0,0,0,0,0,1,6,21,56,126,252,463,804,1365,2366,4368,8736,18565,40410,

%T 87381,184604,379050,758100,1486675,2884776,5592405,10919090,21572460,

%U 43144920,87087001,176565486,357913941,723002336,1453179126,2906358252,5791193143

%N p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^6.

%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).

%C See A291000 for a guide to related sequences.

%H Clark Kimberling, <a href="/A290993/b290993.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6).

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) for n>5. Corrected by _Colin Barker_, Aug 24 2017

%F G.f.: x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)). - _Colin Barker_, Aug 24 2017

%F a(n) = A192080(n-5) for n > 5. - _Georg Fischer_, Oct 23 2018

%F G.f.: x^5/((1-x)^6 - x^6). - _G. C. Greubel_, Apr 11 2023

%p seq(coeff(series(x^5/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Oct 23 2018

%t z = 60; s = x/(1 - x); p = 1 - s^6;

%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000012 *)

%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290993 *)

%o (PARI) concat(vector(5), Vec(x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)) + O(x^50))) \\ _Colin Barker_, Aug 24 2017

%o (GAP) a:=[0,0,0,0,1];; for n in [6..35] do a[n]:=6*a[n-1]-15*a[n-2]+20*a[n-3]-15*a[n-4]+6*a[n-5]; od; Concatenation([0],a); # _Muniru A Asiru_, Oct 23 2018

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0] cat Coefficients(R!( x^5/((1-x)^6 - x^6) )); // _G. C. Greubel_, Apr 11 2023

%o (SageMath)

%o def A290993_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( x^5/((1-x)^6 - x^6) ).list()

%o A290993_list(60) # _G. C. Greubel_, Apr 11 2023

%Y Cf. A000012, A033453, A192080, A289780, A290991, A290992, A291000.

%Y 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), this sequence (m=6), A290994 (m=7), A290995 (m=8).

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Aug 21 2017