A290895 p-INVERT of the positive integers, where p(S) = 1 - S^7.

0, 0, 0, 0, 0, 0, 1, 14, 105, 560, 2380, 8568, 27132, 77521, 203518, 497826, 1148126, 2527609, 5401676, 11508168, 25437917, 60978022, 162008098, 468103230, 1409724358, 4259541790, 12617126893, 36241765553, 100599743538, 269998374114, 702694008002

Offset: 0

Author

Clark Kimberling, Aug 15 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 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 (14, -91, 364, -1001, 2002, -3003, 3433, -3003, 2002, -1001, 364, -91, 14, -1)

Crossrefs

Cf. A000027, A290890.

Programs

Mathematica

z = 60; s = x/(1 - x)^2; p = 1 - s^7;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290895 *)

PARI

concat(vector(6), Vec(x^6 / ((1 - 3*x + x^2)*(1 - 11*x + 57*x^2 - 182*x^3 + 398*x^4 - 626*x^5 + 727*x^6 - 626*x^7 + 398*x^8 - 182*x^9 + 57*x^10 - 11*x^11 + x^12)) + O(x^50))) \\ Colin Barker, Aug 16 2017

Formula

a(n) = 14*a(n-1) - 91*a(n-2) + 364*a(n-3) - 1001*a(n-4) + 2002*a(n-5) - 3003*a(n-6) + 3433*a(n-7) - 3003*a(n-8) + 2002*a(n-9) - 1001*a(n-10) + 364*a(n-11) - 91*a(n-12) + 14*a(n-13) - a(n-14).

G.f.: x^6 / ((1 - 3*x + x^2)*(1 - 11*x + 57*x^2 - 182*x^3 + 398*x^4 - 626*x^5 + 727*x^6 - 626*x^7 + 398*x^8 - 182*x^9 + 57*x^10 - 11*x^11 + x^12)). - Colin Barker, Aug 16 2017