OFFSET
0,1
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).
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:
p(S) t(1,1,0,0,0,...)
1 - S A000045 (Fibonacci numbers)
1 - S^2 A094686
1 - S^3 A115055
1 - S^4 A291379
1 - S^5 A281380
1 - S^6 A281381
1 - 2 S A002605
1 - 3 S A125145
(1 - S)^2 A001629
(1 - S)^3 A001628
(1 - S)^4 A001629
(1 - S)^5 A001873
(1 - S)^6 A001874
1 - S - S^2 A123392
1 - 2 S - S^2 A291382
1 - S - 2 S^2 A124861
1 - 2 S - S^2 A291383
(1 - 2 S)^2 A073388
(1 - 3 S)^2 A291387
(1 - 5 S)^2 A291389
(1 - 6 S)^2 A291391
(1 - S)(1 - 2 S) A291393
(1 - S)(1 - 3 S) A291394
(1 - 2 S)(1 - 3 S) A291395
(1 - S)(1 - 2 S) A291393
(1 - S)(1 - 2 S)(1 - 3 S) A291396
1 - S - S^3 A291397
1 - S^2 - S^3 A291398
1 - S - S^2 - S^3 A186812
1 - S - S^2 - S^3 - S^4 A291399
1 - S^2 - S^4 A291400
1 - S - S^4 A291401
1 - S^3 - S^4 A291402
1 - 2 S^2 - S^4 A291403
1 - S^2 - 2 S^4 A291404
1 - 2 S^2 - 2 S^4 A291405
1 - S^3 - S^6 A291407
(1 - S)(1 - S^2) A291408
(1 - S^2)(1 - S)^2 A291409
1 - S - S^2 - 2 S^3 A291410
1 - 2 S - S^2 + S^3 A291411
1 - S - 2 S^2 + S^3 A291412
1 - 3 S + S^2 + S^3 A291413
1 - 2 S + S^3 A291414
1 - 3 S + S^2 A291415
1 - 4 S + S^2 A291416
1 - 4 S + 2 S^2 A291417
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 3, 2, 1)
FORMULA
G.f.: (-2 - 3 x - 2 x^2 - x^3)/(-1 + 2 x + 3 x^2 + 2 x^3 + x^4).
a(n) = 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) for n >= 5.
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 04 2017
STATUS
approved