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%I #10 Sep 08 2022 08:46:19
%S 0,1,6,20,56,160,480,1456,4384,13136,39360,118064,354272,1062928,
%T 3188736,9565936,28697632,86093264,258280512,774841520,2324523104,
%U 6973567888,20920705152,62762119792,188286360736,564859074896,1694577214656,5083731648560
%N p-INVERT of the odd positive integers, where p(S) = 1 - S^2.
%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 In the following guide to p-INVERT sequences using s = (1,3,5,7,9,...) = A005408, in some cases t(1,3,5,7,9,...) is a shifted (or differently indexed) version of the cited sequence:
%C p(S) *********** t(1,3,5,7,9,...)
%C 1 - S A003946
%C 1 - S^2 A292480
%C 1 - S^3 (not yet in OEIS)
%C (1 - S)^2 (not yet in OEIS)
%C (1 - S)^3 (not yet in OEIS)
%C 1 - S - S^2 A289786
%C 1 + S - S^2 A289484
%C 1 - S - 2 S^2 A289785
%C 1 - S - 3 S^2 A289786
%C 1 - S - 4 S^2 A289787
%C 1 - S - 5 S^2 A289788
%C 1 - S - 6 S^2 A289789
%C 1 - S - 7 S^2 A289790
%C 1 + S - 2 S^2 A289791
%C 1 - S + S^2 - S^3 A289792
%C 1 + S - 3 S^2 A289793
%C 1 - S - S^2 - S^3 A289794
%H Clark Kimberling, <a href="/A292480/b292480.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,6)
%F G.f.: x*(1 + x)^2/((1 - 3*x)*(1 - x + 2*x^2)).
%F a(n) = 4*a(n-1) - 5*a(n-2) + 6*a(n-3) for n >= 5.
%e s = (1,3,5,7,9,...), S(x) = x + 3 x^2 + 5 x^3 + 7 x^4 + ...,
%e p(S(x)) = 1 - ( x + 3 x^2 + 5 x^3 + 7 x^4 + ...)^2,
%e 1/p(S(x)) = 1 + x^2 + 6 x^3 + 20 x^4 + 56 x^5 + ...
%e T(x) = (-1 + 1/p(S(x)))/x = x + 6 x^2 + 20 x^3 + 56 x^4 + ...
%e t(s) = (0,1,2,20,56,...).
%t z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s^2;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292480 *)
%t Join[{0}, LinearRecurrence[{4, -5, 6}, {1, 6, 20}, 30]] (* _Vincenzo Librandi_, Oct 03 2017 *)
%o (Magma) I:=[0,1,6,20]; [n le 4 select I[n] else 4*Self(n-1)- 5*Self(n-2)+6*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Oct 03 2017
%Y Cf. A005408, A292479.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Oct 02 2017
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