A290996 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^4.

1, 2, 4, 9, 22, 55, 136, 330, 789, 1872, 4433, 10510, 24968, 59409, 141470, 336935, 802340, 1910166, 4546845, 10822176, 25758097, 61308650, 145928764, 347350473, 826795942, 1968018151, 4684451824, 11150316882, 26540849109, 63174538224, 150372815489

Offset: 0

Author

Clark Kimberling, Aug 22 2017

Keywords

• nonn
• easy

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 A291000 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 (5,-9,7,-1).

Crossrefs

Cf. A000012, A033453, A289780, A291000.

Programs

Magma

I:=[1,2,4,9]; [n le 4 select I[n] else 5*Self(n-1) -9*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 13 2023

Mathematica

z = 60; s = x/(1-x); p = 1 -s -s^4;
Drop[CoefficientList[Series[s, {x,0,z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290996 *)
LinearRecurrence[{5,-9,7,-1}, {1,2,4,9}, 60] (* G. C. Greubel, Apr 13 2023 *)

PARI

Vec((1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4) + O(x^50)) \\ Colin Barker, Aug 22 2017

SageMath

@CachedFunction
def a(n): # a = A290996
    if(n<4): return (1,2,4,9)[n]
    else: return 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4)
[a(n) for n in range(61)] # G. C. Greubel, Apr 13 2023

Formula

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4) for n >= 4.

G.f.: (1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4). - Colin Barker, Aug 22 2017