A292480 -id:A292480 - cp's OEIS frontend

A292489 p-INVERT of the odd positive integers, where p(S) = 1 - S - 6 S^2.

1, 10, 60, 312, 1656, 8928, 48024, 257904, 1385352, 7442784, 39985272, 214811280, 1154025000, 6199749504, 33306803352, 178933509936, 961281138888, 5164272731808, 27743925989304, 149048175357648, 800728728609384, 4301739993919680, 23110157427289560

Offset: 0

Clark Kimberling, Oct 03 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 A292480 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 (5, -1, 15, 6)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - 6 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292489 *)

x='x+O('x^99); Vec(((1+x)*(1+4*x+7*x^2))/((1-5*x-2*x^2)*(1+3*x^2))) \\ Altug Alkan, Oct 03 2017

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

a(n) = 5*a(n-1) - a(n-2) + 16*a(n-3) + 6*a(n-4) for n >= 5.

A292490 p-INVERT of the odd positive integers, where p(S) = 1 - S - 7 S^2.

Original entry on oeis.org

1, 11, 68, 365, 2019, 11328, 63321, 353483, 1974124, 11026373, 61584323, 343956104, 1921047729, 10729356747, 59925127764, 334691142941, 1869302113507, 10440343236752, 58310941508105, 325675681470731, 1818949357172988, 10159115194159989, 56740239146359107

Offset: 0

Clark Kimberling, Oct 03 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 A292480 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 (5, 0, 17, 7)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - 7 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292490 *)

x='x+O('x^99); Vec(((1+x)*(1+5*x+8*x^2))/(1-5*x-17*x^3-7*x^4)) \\ Altug Alkan, Oct 03 2017

G.f.: -(((1 + x) (1 + 5 x + 8 x^2))/(-1 + 5 x + 17 x^3 + 7 x^4)).

a(n) = 5*a(n-1) + 17*a(n-3) + 7*a(n-4) for n >= 5.

A292491 p-INVERT of the odd positive integers, where p(S) = 1 + S - 2 S^2.

Original entry on oeis.org

-1, 0, 8, 16, 24, 96, 360, 1008, 2808, 8640, 26568, 79056, 235224, 707616, 2128680, 6380208, 19123128, 57386880, 172213128, 516586896, 1549603224, 4648967136, 13947373800, 41841649008, 125523529848, 376572006720, 1129720271688, 3389156563536, 10167456936024

Offset: 0

Clark Kimberling, Oct 03 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 A292480 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 (3, -3, 9)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 + s - 2 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292491 *)

x='x+O('x^99); Vec(((1+x)*(-1+4*x+x^2))/((1-3*x)*(1+3*x^2))) \\ Altug Alkan, Oct 03 2017

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

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

A292492 p-INVERT of the odd positive integers, where p(S) = 1 - S + S^2 - S^3.

Original entry on oeis.org

1, 3, 5, 8, 22, 100, 444, 1680, 5496, 16096, 43936, 117360, 323056, 946288, 2930320, 9287792, 29222800, 89856944, 269619792, 795460592, 2334102160, 6882700336, 20508738256, 61728245104, 186833742864, 565643533232, 1706639551568, 5125652284144, 15338915301264

Offset: 0

Clark Kimberling, Oct 03 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 A292480 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, -19, 23, -8, 6)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s + s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292492 *)

x='x+O('x^99); Vec(((1+x)*(1-5*x+8*x^2-x^3+x^4))/((1-3*x)*(1-4*x+7*x^2-2*x^3+2*x^4))) \\ Altug Alkan, Oct 03 2017

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

a(n) = 7*a(n-1) - 19*a(n-2) + 23*a(n-3) - 8*a(n-4) + 6*a(n-5) for n >= 7.

A292493 p-INVERT of the odd positive integers, where p(S) = 1 + S - 3 S^2.

Original entry on oeis.org

-1, 1, 12, 25, 61, 266, 963, 3053, 10220, 35413, 120345, 405682, 1376119, 4676201, 15859212, 53768225, 182400581, 618792826, 2098887003, 7119249973, 24149097580, 81915342653, 277858469505, 942504046562, 3197013067439, 10844389616401, 36784545696012

Offset: 0

Clark Kimberling, Oct 04 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 A292480 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 (3, -2, 11, 1)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 + s + 3 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292493 *)

x='x+O('x^99); Vec(((1+x)*(1-5*x-2*x^2))/(-1+3*x-2*x^2+11*x^3+x^4)) \\ Altug Alkan, Oct 05 2017

G.f.: -(((1 + x) (-1 + 5 x + 2 x^2))/(-1 + 3 x - 2 x^2 + 11 x^3 + x^4)).

a(n) = 3*a(n-1) - 2*a(n-2) + 11*a(n-3) + a(n-4) for n >= 5.

A292494 p-INVERT of the odd positive integers, where p(S) = 1 - S - S^2 - S^3.

Original entry on oeis.org

1, 5, 21, 88, 362, 1470, 5940, 23996, 97028, 392592, 1588840, 6430088, 26021472, 105301184, 426118816, 1724362608, 6977946160, 28237566352, 114268643984, 462409605552, 1871227376592, 7572272759344, 30642622403664, 124001121308400, 501793808163600

Offset: 0

Clark Kimberling, Oct 04 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 A292480 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, -17, 23, -12, 6, 2)

Cf. A005408, A292480.

z = 60; s = x (x + 1)/(1 - x)^2; p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292494 *)

G.f.: -(((1 + x) (1 - 3 x + 6 x^2 - 3 x^3 + 3 x^4))/(-1 + 7 x - 17 x^2 + 23 x^3 - 12 x^4 + 6 x^5 + 2 x^6)).

a(n) = 7*a(n-1) - 17*a(n-2) + 23*a(n-3) + 12*a(n-4) + 6*a(n-5) + 2*a(n-6) for n >= 7.

cp's OEIS Frontend

A292489 p-INVERT of the odd positive integers, where p(S) = 1 - S - 6 S^2.

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A292490 p-INVERT of the odd positive integers, where p(S) = 1 - S - 7 S^2.

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A292491 p-INVERT of the odd positive integers, where p(S) = 1 + S - 2 S^2.

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A292492 p-INVERT of the odd positive integers, where p(S) = 1 - S + S^2 - S^3.

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PARI

Formula

A292493 p-INVERT of the odd positive integers, where p(S) = 1 + S - 3 S^2.

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Crossrefs

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Mathematica

PARI

Formula

A292494 p-INVERT of the odd positive integers, where p(S) = 1 - S - S^2 - S^3.

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Author

Keywords

Comments

Links

Crossrefs

Programs

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Formula