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%I #13 Oct 04 2017 11:15:37
%S 2,7,22,70,222,705,2238,7105,22556,71608,227332,721705,2291178,
%T 7273743,23091762,73308814,232731578,738846865,2345597854,7446508273,
%U 23640235416,75050038224,238259397096,756395887969,2401310279090,7623377054503,24201736119310
%N p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - 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 = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:
%C p(S) t(1,1,0,0,0,...)
%C 1 - S A000045 (Fibonacci numbers)
%C 1 - S^2 A094686
%C 1 - S^3 A115055
%C 1 - S^4 A291379
%C 1 - S^5 A281380
%C 1 - S^6 A281381
%C 1 - 2 S A002605
%C 1 - 3 S A125145
%C (1 - S)^2 A001629
%C (1 - S)^3 A001628
%C (1 - S)^4 A001629
%C (1 - S)^5 A001873
%C (1 - S)^6 A001874
%C 1 - S - S^2 A123392
%C 1 - 2 S - S^2 A291382
%C 1 - S - 2 S^2 A124861
%C 1 - 2 S - S^2 A291383
%C (1 - 2 S)^2 A073388
%C (1 - 3 S)^2 A291387
%C (1 - 5 S)^2 A291389
%C (1 - 6 S)^2 A291391
%C (1 - S)(1 - 2 S) A291393
%C (1 - S)(1 - 3 S) A291394
%C (1 - 2 S)(1 - 3 S) A291395
%C (1 - S)(1 - 2 S) A291393
%C (1 - S)(1 - 2 S)(1 - 3 S) A291396
%C 1 - S - S^3 A291397
%C 1 - S^2 - S^3 A291398
%C 1 - S - S^2 - S^3 A186812
%C 1 - S - S^2 - S^3 - S^4 A291399
%C 1 - S^2 - S^4 A291400
%C 1 - S - S^4 A291401
%C 1 - S^3 - S^4 A291402
%C 1 - 2 S^2 - S^4 A291403
%C 1 - S^2 - 2 S^4 A291404
%C 1 - 2 S^2 - 2 S^4 A291405
%C 1 - S^3 - S^6 A291407
%C (1 - S)(1 - S^2) A291408
%C (1 - S^2)(1 - S)^2 A291409
%C 1 - S - S^2 - 2 S^3 A291410
%C 1 - 2 S - S^2 + S^3 A291411
%C 1 - S - 2 S^2 + S^3 A291412
%C 1 - 3 S + S^2 + S^3 A291413
%C 1 - 2 S + S^3 A291414
%C 1 - 3 S + S^2 A291415
%C 1 - 4 S + S^2 A291416
%C 1 - 4 S + 2 S^2 A291417
%H Clark Kimberling, <a href="/A291382/b291382.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 3, 2, 1)
%F G.f.: (-2 - 3 x - 2 x^2 - x^3)/(-1 + 2 x + 3 x^2 + 2 x^3 + x^4).
%F a(n) = 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) for n >= 5.
%t z = 60; s = x + x^2; p = 1 - 2 s - s^2;
%t Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A019590 *)
%t Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291382 *)
%Y Cf. A019590, A290890, A291000, A291219, A291728, A292479, A292480.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Sep 04 2017
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