A006645 -id:A006645 - cp's OEIS frontend

A292327 p-INVERT of the Fibonacci sequence (A000045), where p(S) = (1 - S)^2.

2, 5, 14, 38, 102, 271, 714, 1868, 4858, 12569, 32374, 83058, 212350, 541219, 1375570, 3487384, 8821170, 22266413, 56098206, 141087934, 354268502, 888238903, 2223968666, 5561234916, 13889778218, 34652529473, 86361653126, 215021205770, 534861620718

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 (4,-2,-4,-1).

Cf. A000045, A292328.

Cf. A006645, A000129.

z = 60; s = x/(1 - x - x^2); p = (1 - s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000045 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292327 *)

G.f.: -(2 + x)*(-1 + 2*x)/(-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) = A006645(n+1) +2*A000129(n+1). - R. J. Mathar, Jul 08 2022

A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 12, 28, 20, 4, 29, 88, 94, 40, 5, 70, 262, 372, 244, 70, 6, 169, 752, 1333, 1184, 539, 112, 7, 408, 2104, 4472, 5016, 3144, 1064, 168, 8, 985, 5776, 14316, 19408, 15526, 7344, 1932, 240, 9

Offset: 0

Philippe Deléham, Dec 07 2011

Diagonal sums: A201967(n), row sums: A000302(n) (powers of 4).

			Triangle begins:
    1;
    2,   2;
    5,   8,    3;
   12,  28,   20,    4;
   29,  88,   94,   40,   5;
   70, 262,  372,  244,  70,   6;
  169, 752, 1333, 1184, 539, 112, 7;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000027, A000129, A006645, A007290.

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k<0 or  k>n  then 0
    else 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

With[{m = 8}, CoefficientList[CoefficientList[Series[1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k) = if(nMichel Marcus, Feb 17 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0 and n==0): return 1
    else: return 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020

G.f.: 1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A000302(n), A138395(n), A057084(n) for x = -1, 0, 1, 2, 3, respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000027(n), A000302(n), A090018(n), A057091(n) for x = 0, 1, 2, 3, respectively.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

a(40) corrected by Georg Fischer, Feb 17 2020

A261056 Expansion of ( 2-x^2 ) / (x^2+2*x-1)^2 .

Original entry on oeis.org

2, 8, 27, 84, 248, 708, 1973, 5400, 14574, 38896, 102863, 269964, 703972, 1825612, 4711785, 12110064, 31010266, 79148184, 201420163, 511233156, 1294489296, 3270662036, 8247316765, 20758752648, 52163239622, 130875524096, 327893289783, 820410399804, 2050189683644

Offset: 0

R. J. Mathar, Aug 08 2015

2nd column of A210637.

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1)

Cf. A210637.

CoefficientList[Series[(2-x^2)/(x^2+2*x-1)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-2,-4,-1},{2,8,27,84},40] (* Harvey P. Dale, Feb 24 2017 *)

a(n) = 2*A006645(n+2) -A006645(n).

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

Original entry on oeis.org

1, 1, -4, -32, 400, 6912, -153664, -4194304, 136048896, 5120000000, -219503494144, -10567230160896, 564668382613504, 33174037869887488, -2125764000000000000, -147573952589676412928, 11034809241396899282944, 884295678882933431599104, -75613185918270483380568064

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Fibonacci polynomials.

Fibonacci polynomials are defined as F(0)=0, F(1)=1 and F(n)=x*F(n-1)+F(n-2) for n>1. Coefficients are given in triangle A168561 with offset 1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A168561, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A127670.

[(-1)^((n-2)*(n-1) div 2)*2^(n-1)*n^(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 27 2018

Array[(-1)^((#-2)*(#-1)/2)*2^(#-1)*#^(#-3)&,20]

concat([1], [poldisc(p) | p<-Vec(x/(1-x^2-y*x) - x + O(x^20))]) \\ Andrew Howroyd, Aug 26 2018

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).

Original entry on oeis.org

1, 1, -16, -2048, 1638400, 7247757312, -164995463643136, -18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, -271732164163901599116133024293512544256, -13717048991958695477963985711266803110069141504, 3074347100178259797134292590832254504315406543889629184

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Pell polynomials.

Pell polynomials are defined as P(0)=0, P(1)=1 and P(n)=2xP(n-1)+P(n-2) for n>1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Pell Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A086804.

Array[(-1)^((#-2)*(#-1)/2)* 2^((#-1)^2)*#^(#-3)&,15]

cp's OEIS Frontend

A292327 p-INVERT of the Fibonacci sequence (A000045), where p(S) = (1 - S)^2.

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A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A261056 Expansion of ( 2-x^2 ) / (x^2+2*x-1)^2 .

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A317403 a(n)=(-1)^((n-2)*(n-1)/2)*2^(n-1)*n^(n-3).

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A317450 a(n)=(-1)^((n-2)*(n-1)/2)*2^((n-1)^2)*n^(n-3).

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A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).