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A291242
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p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - 2 S - S^2 + S^3.
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2
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2, 5, 13, 35, 91, 241, 631, 1662, 4362, 11470, 30127, 79179, 208023, 546633, 1436257, 3773939, 9916134, 26055432, 68461966, 179888381, 472667065, 1241962303, 3263330095, 8574599917, 22530279167, 59199680826, 155550750026, 408719050346, 1073934109927
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OFFSET
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0,1
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COMMENTS
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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).
See A291219 for a guide to related sequences.
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LINKS
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FORMULA
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G.f.: (-2 - x + 5*x^2 + x^3 - 2*x^4)/(-1 + *x + 4 x^2 - 5*x^3 - 4*x^4 + 2*x^5 + x^6).
a(n) = 2*a(n-1) + 4*a(n-2) - 5*a(n-3) - 4*a(n-4) + 2*a(n-5) + a(n-6) for n >= 7.
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MATHEMATICA
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z = 60; s = x/(1 - x^2); p = 1 - 2 s - s^2 + s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A291242 *)
LinearRecurrence[{2, 4, -5, -4, 2, 1}, {2, 5, 13, 35, 91, 241}, 30] (* Vincenzo Librandi, Aug 29 2017 *)
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PROG
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(Magma) I:=[2, 5, 13, 35, 91, 241]; [n le 6 select I[n] else 2*Self(n-1)+4*Self(n-2)-5*Self(n-3)-4*Self(n-4)+2*Self(n-5)+Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 29 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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