A291000 -id:A291000 - cp's OEIS frontend

A290997 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.

0, 0, 1, 3, 6, 12, 27, 63, 143, 315, 684, 1479, 3195, 6903, 14932, 32361, 70266, 152775, 332397, 723330, 1573829, 3423444, 7444722, 16185939, 35185779, 76483890, 166253545, 361396431, 785621808, 1707884880, 3712912632, 8071922817, 17548551692, 38150905170

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 (6,-15,21,-18,9,-1).

Cf. A000012, A033453, A289780, A291000.

R:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) )); // G. C. Greubel, Apr 14 2023

z = 60; s = x/(1-x); p= 1 -s^3 -s^6;
Drop[CoefficientList[Series[s, {x,0,z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x,0,z}], x], 1]  (* A290997 *)
LinearRecurrence[{6,-15,21,-18,9,-1}, {0,0,1,3,6,12}, 40] (* G. C. Greubel, Apr 14 2023 *)

concat(vector(2), Vec(x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) + O(x^50))) \\ Colin Barker, Aug 22 2017

def A290997_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1-3*x+3*x^2)/(1-6*x+15*x^2-21*x^3 + 18*x^4-9*x^5+x^6) ).list()
A290997_list(40) # G. C. Greubel, Apr 14 2023

a(n) = 6*a(n-1) - 15*a(n-2) + 21*a(n-3) - 18*a(n-4) + 9*a(n-5) - a(n-6) for n >= 7.

G.f.: x^2*(1 - 3*x + 3*x^2) / (1 - 6*x + 15*x^2 - 21*x^3 + 18*x^4 - 9*x^5 + x^6). - Colin Barker, Aug 22 2017

A116703 Number of permutations of length n which avoid the patterns 231, 4123.

Original entry on oeis.org

1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945, 677894, 1671393, 4120937, 10160465, 25051354, 61765902, 152288233, 375477484, 925766477, 2282543187, 5627772815, 13875674756, 34211464510, 84350802705

Offset: 1

Lara Pudwell, Feb 26 2006

Also number of permutations of length n which avoid the patterns 312, 2341, 3412; or avoid the patterns 132, 1324, 3214, etc.

Except for the offset, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^3; see A291000. - Clark Kimberling, Aug 24 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, The Electronic Journal of Combinatorics, Volume 22, Issue 1 (2015) Paper #P1.58.
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
David Callan, Toufik Mansour, Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97.
Toufik Mansour and Mark Shattuck, Avoidance of type (1,2) patterns by Catalan words, Turkish Journal of Analysis and Number Theory, May 2017. See item 1-23 in Table 1, p. 3.
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
L. Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015.
L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, Slides, Permutation Patterns 2014, East Tennessee State University Jul 07 2014.
Index entries for linear recurrences with constant coefficients, signature (4,-5,3).

Cf. A000930.

CoefficientList[Series[x*(1-2*x+2*x^2)/(1-4*x+5*x^2-3*x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 29 2017 *)

x='x+O('x^50); Vec(x*(1-2*x+2*x^2)/(1-4*x+5*x^2-3*x^3)) \\ G. C. Greubel, Apr 29 2017

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

Binomial transform of A000930 starting with offset 1: [1, 1, 2, 3, 4, 6, 9, ...]. - Gary W. Adamson, Oct 23 2007

Edited by N. J. A. Sloane, Mar 16 2008

A136775 Number of multiplex juggling sequences of length n, base state <1,1> and hand capacity 2.

Original entry on oeis.org

1, 3, 11, 40, 145, 525, 1900, 6875, 24875, 90000, 325625, 1178125, 4262500, 15421875, 55796875, 201875000, 730390625, 2642578125, 9560937500, 34591796875, 125154296875, 452812500000, 1638291015625, 5927392578125, 21445507812500, 77590576171875

Offset: 1

Steve Butler, Jan 21 2008

Except for the initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - 3 S + S^2; see A291000. - Clark Kimberling, Aug 24 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Carolina Benedetti, Christopher R. H. Hanusa, Pamela E. Harris, Alejandro H. Morales, Anthony Simpson, Kostant's partition function and magic multiplex juggling sequences, arXiv:2001.03219 [math.CO], 2020. See Table 1 p. 12.
S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597 [math.CO], 2008.
P. E. Harris, E. Insko, L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055, 2014
Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A136776.

R:=PowerSeriesRing(Integers(), 27); Coefficients(R!( (x-2*x^2+x^3)/(1-5*x+5*x^2))); // Marius A. Burtea, Jan 13 2020

CoefficientList[Series[(x^2-2x+1)/(5x^2-5x+1),{x,0,30}],x] (* Harvey P. Dale, Jun 22 2014 *)

Vec((x-2*x^2+x^3)/(1-5*x+5*x^2) + O(x^30)) \\ Colin Barker, Aug 31 2016

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

a(n) = 5*a(n-1)-5*a(n-2) for n>3. - Colin Barker, Aug 31 2016

A290993 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 804, 1365, 2366, 4368, 8736, 18565, 40410, 87381, 184604, 379050, 758100, 1486675, 2884776, 5592405, 10919090, 21572460, 43144920, 87087001, 176565486, 357913941, 723002336, 1453179126, 2906358252, 5791193143

Offset: 0

Clark Kimberling, Aug 21 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 (6,-15,20,-15,6).

Cf. A000012, A033453, A192080, A289780, A290991, A290992, A291000.

Sequences of the form x^(m-1)/((1-x)^m - x^m): A000079 (m=1), A131577 (m=2), A024495 (m=3), A000749 (m=4), A139761 (m=5), this sequence (m=6), A290994 (m=7), A290995 (m=8).

a:=[0,0,0,0,1];;  for n in [6..35] do a[n]:=6*a[n-1]-15*a[n-2]+20*a[n-3]-15*a[n-4]+6*a[n-5]; od; Concatenation([0],a); # Muniru A Asiru, Oct 23 2018

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0] cat Coefficients(R!( x^5/((1-x)^6 - x^6) )); // G. C. Greubel, Apr 11 2023

seq(coeff(series(x^5/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018

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]  (* A290993 *)

concat(vector(5), Vec(x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)) + O(x^50))) \\ Colin Barker, Aug 24 2017

def A290993_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5/((1-x)^6 - x^6) ).list()
A290993_list(60) # G. C. Greubel, Apr 11 2023

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) for n>5. Corrected by Colin Barker, Aug 24 2017
G.f.: x^5 / ((1 - 2*x)*(1 - x + x^2)*(1 - 3*x + 3*x^2)). - Colin Barker, Aug 24 2017
a(n) = A192080(n-5) for n > 5. - Georg Fischer, Oct 23 2018
G.f.: x^5/((1-x)^6 - x^6). - G. C. Greubel, Apr 11 2023

A290994 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1717, 3017, 5110, 8568, 14756, 27132, 54264, 116281, 257775, 572264, 1246784, 2641366, 5430530, 10861060, 21242341, 40927033, 78354346, 150402700, 291693136, 574274008, 1148548016, 2326683921, 4749439975

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 (7,-21,35,-35,21,-7,2).

Cf. A000012, A033453, A049017, A289780, A291000.

Sequences of the form x^(m-1)/((1-x)^m - x^m): A000079 (m=1), A131577 (m=2), A024495 (m=3), A000749 (m=4), A139761 (m=5), A290993 (m=6), this sequence (m=7), A290995 (m=8).

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/((1-x)^7 - x^7) )); // G. C. Greubel, Apr 11 2023

z = 60; s = x/(1 - x); p = 1 - s^7;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A290994 *)

concat(vector(6), Vec(x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)) + O(x^50))) \\ Colin Barker, Aug 22 2017

def A290994_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6/((1-x)^7 - x^7) ).list()
A290994_list(60) # G. C. Greubel, Apr 11 2023

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + 2*a(n-7) for n >= 8.
G.f.: x^6 / ((1 - 2*x)*(1 - 5*x + 11*x^2 - 13*x^3 + 9*x^4 - 3*x^5 + x^6)). - Colin Barker, Aug 22 2017
a(n) = A049017(n-6) for n > 5. - Georg Fischer, Oct 23 2018
G.f.: x^6/((1-x)^7 - x^7). - G. C. Greubel, Apr 11 2023

A291008 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 7*S^2.

Original entry on oeis.org

0, 7, 14, 70, 224, 868, 3080, 11368, 41216, 150640, 548576, 2000992, 7293440, 26592832, 96946304, 353449600, 1288577024, 4697851648, 17127165440, 62441440768, 227645874176, 829940392960, 3025756030976, 11031154419712, 40216845025280, 146620616568832

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 (2,6).

Cf. A000012, A083099, A289780, A291000.

[n le 2 select 7*(n-1) else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 01 2023

z = 60; s = x/(1 - x); p = 1 - s^7;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291008 *)
LinearRecurrence[{2,6}, {0,7}, 40] (* G. C. Greubel, Jun 01 2023 *)

A291008=BinaryRecurrenceSequence(2,6,0,7)
[A291008(n) for n in range(41)] # G. C. Greubel, Jun 01 2023

G.f.: 7*x/(1 - 2*x - 6*x^2).
a(n) = 2*a(n-1) + 6*a(n-2) for n >= 3.
a(n) = 7*A083099(n).
a(n) = (sqrt(7)*((1+sqrt(7))^n - (1-sqrt(7))^n)) / 2. - Colin Barker, Aug 23 2017
a(n) = 7*i^(n-1)*6^((n-1)/2)*ChebyshevU(n-1, -i/sqrt(6)). - G. C. Greubel, Jun 01 2023

A291012 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^2)(1 - 2S).

Original entry on oeis.org

2, 7, 22, 68, 208, 632, 1912, 5768, 17368, 52232, 156952, 471368, 1415128, 4247432, 12746392, 38247368, 114758488, 344308232, 1032990232, 3099101768, 9297567448, 27893226632, 83680728472, 251044282568, 753137042008, 2259419514632, 6778275321112

Offset: 0

Clark Kimberling, Aug 23 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,-6).

Cf. A000012, A033453, A289780, A291000.

[2] cat [8*3^(n-1) - 2^(n-1): n in [1..40]]; // G. C. Greubel, Jun 04 2023

z = 60; s = x/(1-x); p = (1-s)^2(1-2s);
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* this sequence *)
LinearRecurrence[{5,-6}, {2,7,22}, 40] (* G. C. Greubel, Jun 04 2023 *)

Vec((2 -3*x -x^2)/((1-2*x)*(1-3*x)) + O(x^30)) \\ Colin Barker, Aug 23 2017

[8*3^(n-1) - 2^(n-1) - int(n==0)/6 for n in range(41)] # G. C. Greubel, Jun 04 2023

G.f.: (2 - 3 x - x^2)/(1 - 5*x + 6*x^2).
a(n) = 5*a(n-1) - 6*a(n-2) for n >= 4.
a(n) = (16*3^n - 3*2^n) / 6 for n > 0. - Colin Barker, Aug 23 2017
E.g.f.: (1/6)*(-1 - 3*exp(2*x) + 16*exp(3*x)). - G. C. Greubel, Jun 04 2023

A291024 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - 2 S^2)^2.

Original entry on oeis.org

0, 4, 8, 24, 64, 172, 456, 1200, 3136, 8148, 21064, 54216, 139008, 355196, 904840, 2298720, 5825408, 14729636, 37168008, 93612408, 235369664, 590852172, 1481051720, 3707411472, 9268764096, 23145174388, 57732471752, 143857070376, 358113876352, 890666303260

Offset: 0

Clark Kimberling, Aug 24 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..999
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000012, A033453, A289780, A291000, A291142.

z = 60; s = x/(1 - x); p = 1 - 3 s^2 + 2 s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291024 *)
u/4 (* A291142 *)

concat(0, Vec(4*x*(1 - 2*x) / (1 - 2*x - x^2)^2 + O(x^30))) \\ Colin Barker, Aug 24 2017

G.f.: -((4 (-x + 2 x^2))/(-1 + 2 x + x^2)^2).
a(n) = 4*a(n-1) - 2 a(n-2) - 4*a(n-3) - a(n-4) for n >= 5.
a(n) = 4*A291142(n) for n >= 0.
a(n) = ((1+sqrt(2))^n*(3*sqrt(2) + 2*(-1+sqrt(2))*n) - (1-sqrt(2))^n*(3*sqrt(2) + 2*(1+sqrt(2))*n)) / 4. - Colin Barker, Aug 24 2017
E.g.f.: exp(x)*x*cosh(sqrt(2)*x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jun 07 2025

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

cp's OEIS Frontend

A290997 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^3 - S^6.

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A116703 Number of permutations of length n which avoid the patterns 231, 4123.

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A136775 Number of multiplex juggling sequences of length n, base state <1,1> and hand capacity 2.

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A290993 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^6.

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A290994 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S^7.

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A291008 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 7*S^2.

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A291012 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^2)*(1 - 2*S).

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A291012 p-INVERT of (1,1,1,1,1,...), where p(S) = (1 - S^2)(1 - 2S).