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A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

%I A289780 #37 Apr 06 2020 01:14:19
%S A289780 1,4,14,47,156,517,1714,5684,18851,62520,207349,687676,2280686,
%T A289780 7563923,25085844,83197513,275925586,915110636,3034975799,10065534960,
%U A289780 33382471801,110713382644,367182309614,1217764693607,4038731742156,13394504020957,44423039068114
%N A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.
%C A289780 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).
%C A289780 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 A289780 Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
%C A289780 t(A000012) = t(1,1,1,1,1,1,1,...)    = A001906
%C A289780 t(A000290) = t(1,4,9,16,25,36,...)   = A289779
%C A289780 t(A000027) = t(1,2,3,4,5,6,7,8,...)  = A289780
%C A289780 t(A000045) = t(1,2,3,5,8,13,21,...)  = A289781
%C A289780 t(A000032) = t(2,1,3,4,7,11,14,...)  = A289782
%C A289780 t(A000244) = t(1,3,9,27,81,243,...)  = A289783
%C A289780 t(A000302) = t(1,4,16,64,256,...)    = A289784
%C A289780 t(A000351) = t(1,5,25,125,625,...)   = A289785
%C A289780 t(A005408) = t(1,3,5,7,9,11,13,...)  = A289786
%C A289780 t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
%C A289780 t(A016777) = t(1,4,7,10,13,16,...)   = A289789
%C A289780 t(A016789) = t(2,5,8,11,14,17,...)   = A289790
%C A289780 t(A008585) = t(3,6,9,12,15,18,...)   = A289795
%C A289780 t(A000217) = t(1,3,6,10,15,21,...)   = A289797
%C A289780 t(A000225) = t(1,3,7,15,31,63,...)   = A289798
%C A289780 t(A000578) = t(1,8,27,64,625,...)    = A289799
%C A289780 t(A000984) = t(1,2,6,20,70,252,...)  = A289800
%C A289780 t(A000292) = t(1,4,10,20,35,56,...)  = A289801
%C A289780 t(A002620) = t(1,2,4,6,9,12,16,...)  = A289802
%C A289780 t(A001906) = t(1,3,8,21,55,144,...)  = A289803
%C A289780 t(A001519) = t(1,1,2,5,13,34,...)    = A289804
%C A289780 t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
%C A289780 t(A008619) = t(1,1,2,2,3,3,4,4,...)  = A289806
%C A289780 t(A080513) = t(1,2,2,3,3,4,4,5,...)  = A289807
%C A289780 t(A133622) = t(1,2,1,3,1,4,1,5,...)  = A289809
%C A289780 t(A000108) = t(1,1,2,5,14,42,...)    = A081696
%C A289780 t(A081696) = t(1,1,3,9,29,97,...)    = A289810
%C A289780 t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
%C A289780 t(A175676) = t(1,0,0,2,0,0,3,0,...)  = A289844
%C A289780 t(A079977) = t(1,0,1,0,2,0,3,...)    = A289845
%C A289780 t(A059841) = t(1,0,1,0,1,0,1,...)    = A289846
%C A289780 t(A000040) = t(2,3,5,7,11,13,...)    = A289847
%C A289780 t(A008578) = t(1,2,3,5,7,11,13,...)  = A289828
%C A289780 t(A000142) = t(1!, 2!, 3!, 4!, ...)  = A289924
%C A289780 t(A000201) = t(1,3,4,6,8,9,11,...)   = A289925
%C A289780 t(A001950) = t(2,5,7,10,13,15,...)   = A289926
%C A289780 t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
%C A289780 t(A000045*) = t(0,1,1,2,3,5,...)     = A289975 (* indicates prepended 0's)
%C A289780 t(A000045*) = t(0,0,1,1,2,3,5,...)   = A289976
%C A289780 t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
%C A289780 t(A290990*) = t(0,1,2,3,4,5,...)     = A290990
%C A289780 t(A290990*) = t(0,0,1,2,3,4,5,...)   = A290991
%C A289780 t(A290990*) = t(0,0,01,2,3,4,5,...)  = A290992
%H A289780 Clark Kimberling, <a href="/A289780/b289780.txt">Table of n, a(n) for n = 0..1000</a>
%H A289780 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5, -7, 5, -1)
%F A289780 G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).
%F A289780 a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).
%e A289780 Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
%e A289780 S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
%e A289780 p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
%e A289780 - p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
%e A289780 T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
%e A289780 t(s) = (1,3,8,21,...) = A001906.
%e A289780 ***
%e A289780 Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
%e A289780 S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
%e A289780 p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
%e A289780 - p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
%e A289780 T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
%e A289780 t(s) = (1,4,14,47,...) = A289780.
%t A289780 z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
%t A289780 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
%t A289780 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)
%o A289780 (PARI) x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ _Altug Alkan_, Aug 13 2017
%o A289780 (GAP)
%o A289780 P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # _Muniru A Asiru_, Sep 03 2017
%Y A289780 Cf. A000027.
%K A289780 nonn,easy
%O A289780 0,2
%A A289780 _Clark Kimberling_, Aug 10 2017