A289780 -id:A289780 - cp's OEIS frontend

A289845 p-INVERT of A079977, where p(S) = 1 - S - S^2.

1, 2, 4, 9, 19, 43, 91, 202, 433, 952, 2055, 4494, 9737, 21236, 46099, 100403, 218164, 474833, 1032256, 2245929, 4883690, 10623848, 23103985, 50255443, 109298635, 237734446, 517055409, 1124617945, 2446001258, 5320100761, 11571106298, 25167245524, 54738437517

Offset: 0

Clark Kimberling, Aug 14 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 A289780 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (1, 3, -1, 1, -1, -2, 0, -1)

Cf. A079977, A289780.

z = 60; s = -x/(x^4 + x^2 - 1); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A079977 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (*A289845*)
LinearRecurrence[{1,3,-1,1,-1,-2,0,-1},{1,2,4,9,19,43,91,202},40] (* Harvey P. Dale, Jan 16 2019 *)

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

a(n) = a(n-1) + 3*a(n-2) - a(n-3) + a(n-4) - a(n-5) - 2*a(n-6) - a(n-8).

A289924 p-INVERT of (n!), n >= 1 (A000142, shifted), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 17, 79, 402, 2253, 14037, 98152, 774973, 6911131, 69225314, 771593257, 9470565513, 126755983488, 1834510979193, 28511931874423, 473179672441090, 8346048191981797, 155838573499885229, 3069991622444141848, 63618933765102190149, 1383222300396890185731

Offset: 0

Clark Kimberling, Aug 14 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 A289780 for a guide to related sequences.

Michael De Vlieger, Table of n, a(n) for n = 0..448
Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.

Cf. A000142.

z = 60; s = Sum[k! x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000142 shifted *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289924 *)

A289927 p-INVERT of A014217 (starting at n=1), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 15, 53, 187, 656, 2301, 8071, 28308, 99293, 348275, 1221603, 4284864, 15029495, 52717114, 184909361, 648583888, 2274958177, 7979591823, 27989035739, 98173708464, 344351878525, 1207840857737, 4236595263812, 14860185689435, 52123251095327

Offset: 0

Clark Kimberling, Aug 14 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 A289780 for a guide to related sequences.

Cf. A014217, A289780.

z = 60; r = GoldenRatio; s = Sum[Floor[r^k] x^k, {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A014217 shifted *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289927 *)

Conjectures from Colin Barker, Aug 15 2017: (Start)
G.f.: (1 - x^2 + x^3)*(1 + x - x^3) / (1 - 3*x - 4*x^2 + 7*x^3 + 5*x^4 - 7*x^5 - 4*x^6 + 3*x^7 + x^8).
a(n) = 3*a(n-1) + 4*a(n-2) - 7*a(n-3) - 5*a(n-4) + 7*a(n-5) + 4*a(n-6) - 3*a(n-7) - a(n-8) for n>7.
(End)

A289928 p-INVERT of (1,2,3,5,7,11,13,...); i.e., 1 and the primes (A008578), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 48, 162, 547, 1842, 6206, 20906, 70438, 237326, 799629, 2694199, 9077599, 30585239, 103051135, 347211149, 1169861760, 3941626163, 13280557904, 44746308037, 150764154490, 507971076799, 1711511703373, 5766612400708, 19429501132982, 65464000013233

Offset: 0

Clark Kimberling, Aug 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).

See A289780 for a guide to related sequences.

Cf. A008578, A289847.

z = 60; s = x + Sum[Prime[k] x^(k + 1), {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A008578 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1] (* A289928 *)

A289973 p-INVERT of the lower Wythoff sequence (A000201), where p(S) = 1 - S.

Original entry on oeis.org

1, 4, 11, 33, 96, 280, 818, 2387, 6970, 20347, 59401, 173414, 506261, 1477968, 4314748, 12596384, 36773617, 107356118, 313413177, 914971789, 2671149257, 7798096555, 22765597881, 66461404174, 194026015382, 566435438933, 1653639620681, 4827600476829

Offset: 0

Clark Kimberling, Aug 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).

See A289780 for a guide to related sequences.

Cf. A000201, A289974.

z = 60; r = GoldenRatio; s = Sum[Floor[k*r] x^k, {k, 1, z}]; p = 1 - s;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000201 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289973 *)

A289974 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S.

Original entry on oeis.org

2, 9, 35, 139, 549, 2169, 8571, 33866, 133817, 528755, 2089288, 8255476, 32620147, 128893113, 509299806, 2012413902, 7951720511, 31419907712, 124150565816, 490560415825, 1938368302977, 7659141579267, 30263830481105, 119582517950630, 472510530626342

Offset: 0

Clark Kimberling, Aug 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).

See A289780 for a guide to related sequences.

Cf. A001950, A289973.

z = 60; r = 1 + GoldenRatio; s = Sum[Floor[k*r] x^k, {k, 1, z}]; p = 1 - s;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A001950 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1]  (* A289974 *)

A290996 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^4.

Original entry on oeis.org

1, 2, 4, 9, 22, 55, 136, 330, 789, 1872, 4433, 10510, 24968, 59409, 141470, 336935, 802340, 1910166, 4546845, 10822176, 25758097, 61308650, 145928764, 347350473, 826795942, 1968018151, 4684451824, 11150316882, 26540849109, 63174538224, 150372815489

Offset: 0

Clark Kimberling, Aug 22 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 A291000 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,-9,7,-1).

Cf. A000012, A033453, A289780, A291000.

I:=[1,2,4,9]; [n le 4 select I[n] else 5*Self(n-1) -9*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 13 2023

z = 60; s = x/(1-x); p = 1 -s -s^4;
Drop[CoefficientList[Series[s, {x,0,z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290996 *)
LinearRecurrence[{5,-9,7,-1}, {1,2,4,9}, 60] (* G. C. Greubel, Apr 13 2023 *)

Vec((1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4) + O(x^50)) \\ Colin Barker, Aug 22 2017

@CachedFunction
def a(n): # a = A290996
    if(n<4): return (1,2,4,9)[n]
    else: return 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - a(n-4)
[a(n) for n in range(61)] # G. C. Greubel, Apr 13 2023

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

G.f.: (1 - 3*x + 3*x^2) / (1 - 5*x + 9*x^2 - 7*x^3 + x^4). - Colin Barker, Aug 22 2017

A290999 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 6*S^2.

Original entry on oeis.org

0, 6, 12, 54, 168, 606, 2052, 7134, 24528, 84726, 292092, 1007814, 3476088, 11991246, 41362932, 142682094, 492178848, 1697768166, 5856430572, 20201701974, 69685556808, 240379623486, 829187031012, 2860272179454, 9866479513968, 34034319925206, 117401037420252

Offset: 0

Clark Kimberling, Aug 22 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 A291000 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 (2,5).

Cf. A000012, A002532, A033453, A289780, A290997, A290998, A291000.

[n le 2 select 6*(n-1) else 2*Self(n-1) +5*Self(n-2): n in [1..41]]; // G. C. Greubel, Apr 25 2023

z = 60; s = x/(1 - x); p = 1 - s^6;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290999 *)
LinearRecurrence[{2,5},{0,6},30] (* Harvey P. Dale, Mar 25 2018 *)

A290999=BinaryRecurrenceSequence(2,5,0,6)
[A290999(n) for n in range(41)] # G. C. Greubel, Apr 25 2023

G.f.: 6*x/(1 - 2*x - 5*x^2).
a(n) = 2*a(n-1) + 5*a(n-2) for n >= 3.
a(n) = 6*A002532(n).
a(n) = sqrt(3/2)*((1+sqrt(6))^n - (1-sqrt(6))^n). - Colin Barker, Aug 23 2017

A291001 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 8*S^2.

Original entry on oeis.org

0, 8, 16, 88, 288, 1192, 4400, 17144, 65088, 250184, 955984, 3663256, 14018400, 53679592, 205487984, 786733112, 3011882112, 11530896008, 44144966800, 169006205656, 647027178912, 2477097797416, 9483385847216, 36306456276344, 138996613483200, 532138420900808

Offset: 0

Clark Kimberling, Aug 22 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 A291000 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 (2,7).

Cf. A000012, A015519, A033453, A289780, A291000.

[n le 2 select 8*(n-1) else 2*Self(n-1) +7*Self(n-2): n in [1..41]]; // G. C. Greubel, Apr 25 2023

z = 60; s = x/(1 - x); p = 1 - s^8;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291001 *)
LinearRecurrence[{2,7}, {0,8}, 41] (* G. C. Greubel, Apr 25 2023 *)

A291001=BinaryRecurrenceSequence(2,7,0,8)
[A291001(n) for n in range(41)] # G. C. Greubel, Apr 25 2023

G.f.: 8*x/(1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n >= 3.
a(n) = 8*A015519(n).
a(n) = sqrt(2)*((1+2*sqrt(2))^n - (1-2*sqrt(2))^n). - Colin Barker, Aug 23 2017

A291002 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)(1 - 2S)(1 - 3S).

Original entry on oeis.org

6, 31, 146, 652, 2816, 11896, 49496, 203752, 832376, 3381736, 13683896, 55206952, 222242936, 893219176, 3585623096, 14380739752, 57637717496, 230895178216, 924613703096, 3701553914152, 14815513224056, 59289946122856, 237243465219896, 949224905162152

Offset: 0

Clark Kimberling, Aug 22 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 A291000 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,-26,24).

Cf. A000012, A033453, A289780, A291000.

[(2^n-16*3^n+27*4^n)/2: n in [0..40]]; // G. C. Greubel, Apr 27 2023

z = 60; s = x/(1-x); p = (1-s)*(1-2*s)*(1-3*s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291002 *)
LinearRecurrence[{9,-26,24}, {6,31,146}, 41] (* G. C. Greubel, Apr 27 2023 *)

[(2^n-16*3^n+27*4^n)/2 for n in range(41)] # G. C. Greubel, Apr 27 2023

G.f.: (6 - 23*x + 23*x^2)/(1 - 9*x + 26*x^2 - 24*x^3).
a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3) for n >= 4.
a(n) = (2^n - 16*3^n + 27*4^n) / 2. - Colin Barker, Aug 23 2017
E.g.f.: (1/2)*(exp(2*x) - 16*exp(3*x) + 27*exp(4*x)). - G. C. Greubel, Apr 27 2023

cp's OEIS Frontend

A289845 p-INVERT of A079977, where p(S) = 1 - S - S^2.

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A289924 p-INVERT of (n!), n >= 1 (A000142, shifted), where p(S) = 1 - S - S^2.

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A289927 p-INVERT of A014217 (starting at n=1), where p(S) = 1 - S - S^2.

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A289928 p-INVERT of (1,2,3,5,7,11,13,...); i.e., 1 and the primes (A008578), where p(S) = 1 - S - S^2.

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A289973 p-INVERT of the lower Wythoff sequence (A000201), where p(S) = 1 - S.

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A289974 p-INVERT of the upper Wythoff sequence (A001950), where p(S) = 1 - S.

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A290996 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^4.

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A290999 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 6*S^2.

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A291001 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 8*S^2.

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A291002 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S)(1 - 2S)(1 - 3S).