A290997 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.

0, 0, 1, 3, 6, 12, 27, 63, 143, 315, 684, 1479, 3195, 6903, 14932, 32361, 70266, 152775, 332397, 723330, 1573829, 3423444, 7444722, 16185939, 35185779, 76483890, 166253545, 361396431, 785621808, 1707884880, 3712912632, 8071922817, 17548551692, 38150905170

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 (6,-15,21,-18,9,-1).

Crossrefs

Cf. A000012, A033453, A289780, A291000.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) )); // G. C. Greubel, Apr 14 2023

Mathematica

z = 60; s = x/(1-x); p= 1 -s^3 -s^6;
Drop[CoefficientList[Series[s, {x,0,z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290997 *)
LinearRecurrence[{6,-15,21,-18,9,-1}, {0,0,1,3,6,12}, 40] (* G. C. Greubel, Apr 14 2023 *)

PARI

concat(vector(2), Vec(x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) + O(x^50))) \\ Colin Barker, Aug 22 2017

SageMath

def A290997_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) ).list()
A290997_list(40) # G. C. Greubel, Apr 14 2023

Formula

a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 18*a(n-4) + 9*a(n-5) - a(n-6) for n >= 7.

G.f.: x^2*(1 - 3*x + 3*x^2) / (1 - 6*x + 15*x^2 - 21*x^3 + 18*x^4 - 9*x^5 + x^6). - Colin Barker, Aug 22 2017