A001871 -id:A001871 - cp's OEIS frontend

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

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

A172249 Triangle, read by rows, given by [0,1/3,-1/3,0,0,0,0,0,0,0,...] DELTA [3,-1/3,1/3,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 3, 0, 1, 8, 0, 0, 6, 21, 0, 0, 1, 25, 55, 0, 0, 0, 9, 90, 144, 0, 0, 0, 1, 51, 300, 377, 0, 0, 0, 0, 12, 234, 954, 987, 0, 0, 0, 0, 1, 86, 951, 2939, 2584, 0, 0, 0, 0, 0, 15, 480, 3573, 8850, 6765, 0, 0, 0, 0, 0, 1, 130, 2305, 12707, 26195, 17711, 0, 0, 0, 0, 0, 0, 18, 855

Offset: 0

Philippe Deléham, Jan 29 2010

Diagonal sums : |A077897|. Column sums : A001353 .

			Triangle begins :
1,
0,3,
0,1,8,
0,0,6,21,
0,0,1,25,55,
0,0,0,9,90,144,
0,0,0,1,51,300,377,
0,0,0,0,12,234,954,987,
0,0,0,0,1,86,951,2939,2584,
0,0,0,0,0,15,480,3573,8850,6765,
0,0,0,0,0,1,130,2305,12707,26195,17711,

Cf. A001871, A001906, A125662.

T(n,k):=2*sum((j*binomial(n+j,2*n-2*k+2*j)*binomial(n-k+j,j))/(n+j),j,1,n+k); /* Vladimir Kruchinin_, Oct 28 2020 */

T(n,k) = 3*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k>n or if k<0.
Sum_{k, 0<=k<=n} T(n,k)= 3^n = A000244(n) (row sums).
G.f.: 1/(1-3*x*y-x^2*y+x^2*y^2). - R. J. Mathar, Aug 11 2015
T(n,k) = 2*Sum_{j=1..n+k} j*C(n+j,2*n-2*k+2*j)*C(n-k+j,j)/(n+j), T(0,0)=1. - Vladimir Kruchinin, Oct 28 2020

A286986 Number of connected dominating sets in the n-antiprism graph.

Original entry on oeis.org

3, 15, 54, 175, 543, 1642, 4875, 14271, 41310, 118487, 337263, 953810, 2682579, 7508655, 20929158, 58121407, 160877055, 443993146, 1222110555, 3355879647, 9195143598, 25144855655, 68635721679, 187035899810, 508896450723, 1382653280847, 3751638404310

Offset: 1

Eric W. Weisstein, May 17 2017

Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Connected Dominating Set
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Table[6 n ChebyshevU[n - 1, 3/2] + (1 - 2 n) LucasL[2 n], {n, 30}] (* Eric W. Weisstein, May 17 2017 *)
LinearRecurrence[{6, -11, 6, -1}, {3, 15, 54, 175}, 30] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[(3*x - 3*x^2 - 3*x^3 - 2*x^4)/(1 - 6*x + 11*x^2 - 6*x^3 + x^4), {x,0,50}], x]] (* G. C. Greubel, May 17 2017 *)

x='x+O('x^50); Vec((3*x - 3*x^2 - 3*x^3 - 2*x^4)/(1 - 6*x + 11*x^2 - 6*x^3 + x^4)) \\ G. C. Greubel, May 17 2017

From G. C. Greubel, May 17 2017: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: (3 - 3*x - 3*x^2 - 2*x^3)*x/(1 - 6*x + 11*x^2 - 6*x^3 + x^4). (End)
a(n) = 28*A001871(n) -72*A001871(n-1) -15*A001906(n)-26*A001906(n+1). - R. J. Mathar, Dec 16 2024

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

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

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A172249 Triangle, read by rows, given by [0,1/3,-1/3,0,0,0,0,0,0,0,...] DELTA [3,-1/3,1/3,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A286986 Number of connected dominating sets in the n-antiprism graph.

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Formula

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

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cp's OEIS Frontend

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

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Author

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Magma

Mathematica

PARI

A172249 Triangle, read by rows, given by [0,1/3,-1/3,0,0,0,0,0,0,0,...] DELTA [3,-1/3,1/3,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Author

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Comments

Examples

Crossrefs

Programs

Maxima

Formula

A286986 Number of connected dominating sets in the n-antiprism graph.

Views

Author

Keywords

Links

Programs

Mathematica

PARI

Formula

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

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Mathematica

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).