A292322 -id:A292322 - cp's OEIS frontend

A292324 p-INVERT of (1,0,0,1,0,0,0,0,0,...), where p(S) = (1 - S)^2.

2, 3, 4, 7, 12, 19, 28, 42, 64, 97, 144, 212, 312, 459, 672, 979, 1422, 2062, 2984, 4308, 6206, 8925, 12816, 18376, 26310, 37620, 53728, 76648, 109230, 155507, 221184, 314325, 446320, 633249, 897804, 1271993, 1800942, 2548242, 3603468, 5092747, 7193604

Offset: 0

Clark Kimberling, Sep 15 2017

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).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 2, -2, 0, 0, -1)

Cf. A079978, A292322.

z = 60; s = x + x^4; p = (1 - s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292324 *)

G.f.: -(((-1 + x) (1 + x) (1 - x + x^2) (2 + x + x^2 + x^3))/(-1 + x + x^4)^2).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) - 2*a(n-5) - a(n-8) for n >= 9.
a(n) = a(n-1)+a(n-4)+A003269(n+2). - R. J. Mathar, Mar 19 2024

A292323 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = (1 - S)(1 + S^2).

Original entry on oeis.org

1, 0, 0, 2, 1, 0, 5, 6, 1, 11, 23, 10, 22, 71, 57, 50, 191, 243, 164, 474, 860, 676, 1175, 2674, 2758, 3225, 7626, 10256, 10313, 20882, 34642, 36384, 57921, 108270, 130025, 170606, 321415, 448093, 540825, 934958, 1468860, 1798559, 2750605, 4605556, 6042649

Offset: 0

Clark Kimberling, Sep 15 2017

See A291728 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -1, 4, -2, 1, -3, 1, 0, 1)

Cf. A079978, A292322.

z = 60; s = x/(x - x^3); p = (1 - s)(1 + s^2);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292323 *)

x='x+O('x^99); Vec((1-x+x^2-2*x^3+x^4+x^6)/((1-x-x^3)*(1+x^2-2*x^3+x^6))) \\ Altug Alkan, Oct 05 2017

G.f.: -((1 - x + x^2 - 2 x^3 + x^4 + x^6)/((-1 + x + x^3) (1 + x^2 - 2 x^3 + x^6))).

a(n) = a(n-1) - a(n-2) + 4*a(n-3) - 2*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) + a(n-9) for n >= 10.

A292397 p-INVERT of the tribonacci numbers (A000073(k), k>=2), where p(S) = 1 - S - S^2 - S^3.

Original entry on oeis.org

1, 3, 10, 33, 108, 352, 1144, 3714, 12050, 39084, 126752, 411041, 1332923, 4322363, 14016392, 45451793, 147389276, 477948252, 1549872500, 5025868667, 16297700769, 52849583211, 171378684824, 555740504324, 1802134907175, 5843896942499, 18950374573538

Offset: 0

Clark Kimberling, Sep 18 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -1, -3, -7, 2, 6, 7, 3, 1)

Cf. A000073, A292322.

I:=[1,3,10,33,108,352,1144,3714,12050]; [n le 9 select I[n] else 4*Self(n-1)-Self(n-2)-3*Self(n-3)-7*Self(n-4)+2*Self(n-5)+6*Self(n-6)+7*Self(n-7)+3*Self(n-8)+Self(n-9): n in [1..30]]; // Vincenzo Librandi, Oct 13 2017

z = 60; s = x/(1 - x - x^2 - x^3); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000073 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A292397 *)
LinearRecurrence[{4, -1, -3, -7, 2, 6, 7, 3, 1}, {1, 3, 10, 33, 108, 352, 1144, 3714, 12050}, 30] (* Vincenzo Librandi, Oct 13 2017 *)

x='x+O('x^99); Vec(((1+x+x^2)*(1-2*x+x^3+x^4))/(1-4*x+x^2+3*x^3+7*x^4-2*x^5-6*x^6-7*x^7-3*x^8-x^9)) \\ Altug Alkan, Oct 04 2017

G.f.: -(((1 + x + x^2) (1 - 2 x + x^3 + x^4))/(-1 + 4 x - x^2 - 3 x^3 - 7 x^4 + 2 x^5 + 6 x^6 + 7 x^7 + 3 x^8 + x^9)).

a(n) = 4*a(n-1) - a(n-2) - 3*a(n-3) - 7*a(n-4) + 2*a(n-5) + 6*a(n-6) + 7*a(n-7) + 3*a(n-8) + a(n-9) for n >= 10.

cp's OEIS Frontend

A292324 p-INVERT of (1,0,0,1,0,0,0,0,0,...), where p(S) = (1 - S)^2.

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A292323 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = (1 - S)(1 + S^2).

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A292397 p-INVERT of the tribonacci numbers (A000073(k), k>=2), where p(S) = 1 - S - S^2 - S^3.

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