A290896 p-INVERT of the positive integers, where p(S) = 1 - S^8.

0, 0, 0, 0, 0, 0, 0, 1, 16, 136, 816, 3876, 15504, 54264, 170544, 490315, 1307536, 3269288, 7732144, 17436220, 37819152, 79883544, 167737776, 362063944, 839161648, 2158258904, 6136548496, 18586871324, 57486027952, 176258492200, 527387147664, 1529591016109

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 (16, -120, 560, -1820, 4368, -8008, 11440, -12869, 11440, -8008, 4368, -1820, 560, -120, 16, -1)

Crossrefs

Cf. A000027, A290890.

Programs

Mathematica

z = 60; s = x/(1 - x)^2; p = 1 - s^8;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290896 *)

PARI

concat(vector(7), Vec(x^7 / ((1 - 3*x + x^2)*(1 - x + x^2)*(1 - 4*x + 7*x^2 - 4*x^3 + x^4)*(1 - 8*x + 28*x^2 - 56*x^3 + 71*x^4 - 56*x^5 + 28*x^6 - 8*x^7 + x^8)) + O(x^40))) \\ Colin Barker, Aug 16 2017

Formula

a(n) = 16*a(n-1) - 120*a(n-2) + 560*a(n-3) - 1820*a(n-4) + 4368*a(n-5) - 8008*a(n-6) + 11440*a(n-7) - 12869*a(n-8) + 11440*a(n-9) - 8008*a(n-10) + 4368*a(n-11) - 1820*a(n-12) + 560*a(n-13) - 120*a(n-14) + 16*a(n-15) - a(n-16).

G.f.: x^7 / ((1 - 3*x + x^2)*(1 - x + x^2)*(1 - 4*x + 7*x^2 - 4*x^3 + x^4)*(1 - 8*x + 28*x^2 - 56*x^3 + 71*x^4 - 56*x^5 + 28*x^6 - 8*x^7 + x^8)). - Colin Barker, Aug 16 2017