A137276 -id:A137276 - cp's OEIS frontend

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

-2, -7, -15, -25, -35, -42, -42, -30, 0, 55, 143, 273, 455, 700, 1020, 1428, 1938, 2565, 3325, 4235, 5313, 6578, 8050, 9750, 11700, 13923, 16443, 19285, 22475, 26040, 30008, 34408, 39270, 44625, 50505, 56943, 63973, 71630, 79950, 88970, 98728, 109263, 120615

Offset: 8

Jamel Ghanouchi, Nov 06 2009

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A111373, A135929, A137276.

[Binomial(n-5,3)*(n-16)/4: n in [8..60]]; // G. C. Greubel, Jul 30 2022

Table[(n-5)*(n-6)*(n-7)*(n-16)/24, {n,8,60}] (* G. C. Greubel, Jun 15 2016 *)

[binomial(n-5,3)*(n-16)/4 for n in (8..60)] # G. C. Greubel, Jul 30 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^8*(-2+3*x)/(1-x)^5.
E.g.f.: (1/24)*( -3360 - 1800*x - 420*x^2 - 52*x^3 - 3*x^4 + (3360 - 1560*x + 300*x^2 - 28*x^3 + x^4)*exp(x) ). - G. C. Greubel, Jul 30 2022

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=xp(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = xQ(x, n - 2) - Q(x, n - 3).

Original entry on oeis.org

1, -1, 1, 0, 1, -1, -1, 1, 1, -1, 1, 0, -2, -1, 1, 1, 2, -2, 1, -1, 1, -3, -1, 1, 0, 3, 3, -3, 1, -1, -3, 3, -4, -1, 1, 1, -1, 6, 4, -4, 1

Offset: 1

Roger L. Bagula, Apr 08 2008

Row sums are:
{1, 0, 1, -1, 1, -2, 2, -3, 4, -5, 7}
In differential equation terms this is equivalent to ( in Mathematica notation):
D[y[x],{x,3}]=x*D[y[x],{x,1}]-y[x];
Two simple possible HypergeometricPFQ based results are:
DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 1}, y, x];
DSolve[{D[y[x], {x, 3}] == x*D[y[x], {x, 1}] - y[x], y[0] == 0}, y, x].

			{1},
{-1, 1},
{0, 1},
{-1, -1, 1},
{1, -1, 1},
{0, -2, -1, 1},
{1, 2, -2, 1},
{-1, 1, -3, -1, 1},
{0, 3, 3, -3, 1},
{-1, -3, 3, -4, -1, 1},
{1, -1, 6, 4, -4, 1}

Cf. A000931, A137276.

Clear[Q, x] Q[x, -2] = 1 - x; Q[x, -1] = 0; Q[x, 0] = 1; Q[x_, n_] := Q[x, n] = x*Q[x, n - 2] - Q[x, n - 3]; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Q[x, n], x], {n, 0, 10}]; Flatten[a]

Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).

A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(mt)] of the shifted Boubaker polynomial B_n(2cos t)-2cos(nt).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Haydar Rahmanov, May 05 2009

Starting from the polynomials B_n(x) defined in A137276 and A135929, we insert x=2*cos(t), and define the Fourier coefficients A(n,m) by B_n(2*cos t)-2*cos(n*t) = 4*sum(m=0,..,n-2) A(n,m)*cos(m*t).
A(n,m) is not an integer for n=0, so the table starts at n=1. Furthermore, A(n,m)=0 if n-m is odd, these regular zeros are skipped as usual, so effectively the first table entry appears at n=2.
Simpler definition from R. J. Mathar, Apr 15 2010: a(n)=1 if n =0 or n in A002061, otherwise a(n)=2. So this is a kind of characteristic function of the central polygonal numbers A002061.

			Using T^m =cos(m*t) as a notational shortcut, the expansions start
; B_1(2 cos t)-2*cos 1 t = 0
1 ; B_2(2 cos t)-2*cos 2 t = 1
0 2 ; B_3(2 cos t)-2*cos 3 t = 2*T
1 0 2 ; B_4(2 cos t)-2*cos 4 t = 1+2*T^2
0 2 0 2 ; B_5(2 cos t)-2*cos 5 t = 2*T+2*T^3
1 0 2 0 2 ; B_6(2 cos t)-2*cos 6 t = 1+2*T^2+2*T^4
0 2 0 2 0 2 ; B_7(2 cos t)-2*cos 7 t = 2*T+2*T^3+2*T^5
1 0 2 0 2 0 2 ; B_8(2 cos t)-2*cos 8 t = 1+2*T^2+2*T^4+2*T^6
0 2 0 2 0 2 0 2 ; B_9(2 cos t)-2*cos 9 t = 2*T+2*T^3+2*T^5+2*T^7
1 0 2 0 2 0 2 0 2 ; B_10(2 cos t)-2*cos 10 t = 1+2*T^2+2*T^4+2*T^6+2*T^8
0 2 0 2 0 2 0 2 0 2 ; B_11(2 cos t)-2*cos 11 t = 2*T^3+2*T^5+2*T^7+2*T^9+2*T

A. Luzon and M. A. Morón, Recurrence relations for polynomial sequences via Riordan matrices, arXiv:0904.2672 [math.CO]

centralPolygonalQ[n_] := Resolve[Exists[k, k>0, n == k^2-k+1], Integers];
b[n_] := If[n == 0 || centralPolygonalQ[n], 1, 2];
a[n_] := b[n-1];
Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Oct 31 2018, after R. J. Mathar *)

Definition clarified, publication title corrected, sequence extended by R. J. Mathar, Dec 07 2009

cp's OEIS Frontend

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

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A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=xp(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = xQ(x, n - 2) - Q(x, n - 3).

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A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(mt)] of the shifted Boubaker polynomial B_n(2cos t)-2cos(nt).

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cp's OEIS Frontend

A167543 a(n) = (n-5)*(n-6)*(n-7)*(n-16)/24.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=x*p(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = x*Q(x, n - 2) - Q(x, n - 3).

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A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(m*t)] of the shifted Boubaker polynomial B_n(2*cos t)-2*cos(n*t).

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A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

A137350 A triangular Sequence of coefficients of a three deep polynomial recursion based on a Chebyshev kind and a Padovan recursion: Chebyshev; p(x,n)=xp(x,n-1)-p(x,n-2); Padovan: a(n)=a(n-2)+a(n-3); Q(x, n) = xQ(x, n - 2) - Q(x, n - 3).

A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(mt)] of the shifted Boubaker polynomial B_n(2cos t)-2cos(nt).