login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A291030 p-INVERT of the positive integers, where p(S) = 1 - S - S^2 - S^3 - S^4. 1
1, 4, 15, 56, 208, 767, 2812, 10278, 37530, 137044, 500571, 1828818, 6682264, 24416877, 89218462, 325997507, 1191160160, 4352355633, 15902968338, 58107491971, 212317732888, 775783501558, 2834620130881, 10357363200392, 37844566834330, 138279520124262 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 A290890 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-33,68,-85,68,-33,9,-1).

FORMULA

G.f.: (1 - 5 x + 12 x^2 - 15 x^3 + 12 x^4 - 5 x^5 + x^6)/(1 - 9 x + 33 x^2 - 68 x^3 + 85 x^4 - 68 x^5 + 33 x^6 - 9 x^7 + x^8).

a(n) = 9*a(n-1) - 33*a(n-2) + 68*a(n-3) - 85*a(n-4) + 68*a(n-5) - 33*a(n-6) + 9*a(n-7) - a(n-8).

MATHEMATICA

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2 - s^3 - s^4;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291030 *)

LinearRecurrence[{9, -33, 68, -85, 68, -33, 9, -1}, {1, 4, 15, 56, 208, 767, 2812, 10278}, 40] (* Vincenzo Librandi, Aug 20 2017 *)

PROG

(MAGMA) I:=[1, 4, 15, 56, 208, 767, 2812, 10278]; [n le 8 select I[n] else 9*Self(n-1)-33*Self(n-2)+68*Self(n-3)-85*Self(n-4)+68*Self(n-5)-33*Self(n-6)+9*Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Aug 20 2017

CROSSREFS

Cf. A000027, A290890.

Sequence in context: A082133 A060111 A077824 * A217779 A191606 A242495

Adjacent sequences:  A291027 A291028 A291029 * A291031 A291032 A291033

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 19 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 25 16:42 EST 2020. Contains 331245 sequences. (Running on oeis4.)