A290890 -id:A290890 - cp's OEIS frontend

A291182 p-INVERT of the positive integers, where p(S) = 1 - 3S + 2S^2.

3, 13, 52, 203, 781, 2976, 11267, 42469, 159596, 598499, 2241165, 8383872, 31340691, 117100285, 437378260, 1633244795, 6097779229, 22763575008, 84971451155, 317161317781, 1183776154124, 4418211213011, 16489770106653, 61542705525504, 229685859522339

Offset: 0

Clark Kimberling, Aug 19 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).

See A290890 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 (7,-14,7,-1)

Cf. A000027, A290890.

I:=[3,13,52,203]; [n le 4 select I[n] else 7*Self(n-1)-14*Self(n-2)+7*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 20 2017

z = 60; s = x/(1 - x)^2; p = 1 - 3 s + 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291182 *)
LinearRecurrence[{7, -14, 7, -1}, {3, 13, 52, 203}, 40] (* Vincenzo Librandi, Aug 20 2017 *)

G.f.: (3 - 8 x + 3 x^2)/(1 - 7 x + 14 x^2 - 7 x^3 + x^4).

a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3) - a(n-4).

A291183 p-INVERT of the positive integers, where p(S) = 1 - 4S + 2S^2.

Original entry on oeis.org

4, 22, 116, 608, 3180, 16618, 86812, 453440, 2368292, 12369174, 64601428, 337397536, 1762142540, 9203221994, 48066074172, 251036784256, 1311100720708, 6847542588950, 35762957380148, 186780746599392, 975507894703660, 5094827328491242, 26608975328086364

Offset: 0

Clark Kimberling, Aug 19 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).

See A290890 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 (8,-16,8,-1)

Cf. A000027, A290890.

I:=[4,22,116,608]; [n le 4 select I[n] else 8*Self(n-1)-16*Self(n-2)+8*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 20 2017

z = 60; s = x/(1 - x)^2; p = 1 - 4 s + 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291183 *)
LinearRecurrence[{8, -16, 8, -1}, {4, 22, 116, 608}, 40] (* Vincenzo Librandi, Aug 20 2017 *)

G.f.: (2 (2 - 5 x + 2 x^2))/(1 - 8 x + 16 x^2 - 8 x^3 + x^4).

a(n) = 8*a(n-1) - 16*a(n-2) + 8*a(n-3) - a(n-4).

A291184 p-INVERT of the positive integers, where p(S) = 1 - 4S + 3S^2.

Original entry on oeis.org

4, 21, 104, 507, 2452, 11808, 56732, 272229, 1305400, 6257355, 29988140, 143701056, 688563508, 3299237877, 15807943688, 75741312603, 362900797636, 1738768378464, 8330956025036, 39916050834885, 191249400483544, 916331219497131, 4390407398410844

Offset: 0

Clark Kimberling, Aug 19 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).

See A290890 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 (8, -17, 8, -1)

Cf. A000027, A290890.

z = 60; s = x/(1 - x)^2; p = 1 - 4 s + 3 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291184 *)
LinearRecurrence[{8,-17,8,-1},{4,21,104,507},30] (* Harvey P. Dale, Feb 24 2018 *)

G.f.: (4 - 11 x + 4 x^2)/(1 - 8 x + 17 x^2 - 8 x^3 + x^4).

a(n) = 8*a(n-1) - 17*a(n-2) + 8*a(n-3) - a(n-4).

A291030 p-INVERT of the positive integers, where p(S) = 1 - S - S^2 - S^3 - S^4.

Original entry on oeis.org

1, 4, 15, 56, 208, 767, 2812, 10278, 37530, 137044, 500571, 1828818, 6682264, 24416877, 89218462, 325997507, 1191160160, 4352355633, 15902968338, 58107491971, 212317732888, 775783501558, 2834620130881, 10357363200392, 37844566834330, 138279520124262

Offset: 0

Clark Kimberling, Aug 19 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).

See A290890 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 (9,-33,68,-85,68,-33,9,-1).

Cf. A000027, A290890.

I:=[1,4,15,56,208,767,2812,10278]; [n le 8 select I[n] else 9*Self(n-1)-33*Self(n-2)+68*Self(n-3)-85*Self(n-4)+68*Self(n-5)-33*Self(n-6)+9*Self(n-7)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Aug 20 2017

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2 - s^3 - s^4;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291030 *)
LinearRecurrence[{9, -33, 68, -85, 68, -33, 9, -1}, {1, 4, 15, 56, 208, 767, 2812, 10278}, 40] (* Vincenzo Librandi, Aug 20 2017 *)

G.f.: (1 - 5 x + 12 x^2 - 15 x^3 + 12 x^4 - 5 x^5 + x^6)/(1 - 9 x + 33 x^2 - 68 x^3 + 85 x^4 - 68 x^5 + 33 x^6 - 9 x^7 + x^8).

a(n) = 9*a(n-1) - 33*a(n-2) + 68*a(n-3) - 85*a(n-4) + 68*a(n-5) - 33*a(n-6) + 9*a(n-7) - a(n-8).

cp's OEIS Frontend

A291182 p-INVERT of the positive integers, where p(S) = 1 - 3*S + 2*S^2.

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A291183 p-INVERT of the positive integers, where p(S) = 1 - 4*S + 2*S^2.

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A291184 p-INVERT of the positive integers, where p(S) = 1 - 4*S + 3*S^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A291030 p-INVERT of the positive integers, where p(S) = 1 - S - S^2 - S^3 - S^4.

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Author

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Comments

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Crossrefs

Programs

Magma

Mathematica

Formula

A291182 p-INVERT of the positive integers, where p(S) = 1 - 3S + 2S^2.

A291183 p-INVERT of the positive integers, where p(S) = 1 - 4S + 2S^2.

A291184 p-INVERT of the positive integers, where p(S) = 1 - 4S + 3S^2.