A006974 -id:A006974 - cp's OEIS frontend

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Clark Kimberling, Feb 18 2012

Another version in A201701. - Philippe Deléham, Mar 03 2012
Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012
Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)

Cf. A028297, A207538, A133156.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - Philippe Deléham, Mar 03 2012
G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015
T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021

A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

Original entry on oeis.org

1, 15, 128, 816, 4320, 20064, 84480, 329472, 1208064, 4209920, 14057472, 45260800, 141213696, 428654592, 1270087680, 3683254272, 10478223360, 29297934336, 80648077312, 218864025600, 586290298880, 1551944908800, 4063273943040

Offset: 0

Brad Clardy, Mar 08 2012

The MAGMA program provided calculates the coefficients of order one Chebyshev polynomials, for any arbitrary level. For example, setting Rn to 0 produces A001792, 1 produces A001793, 2 produces A001794, 3 produces A006974, 4 produces A006975, and 5 produces A006976. This sequence is produced with an Rn of 6.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001792, A001793, A001794, A006974, A006975, A006976, A053120.

List([0..30], n-> 2^(n-1)*(n+14)*Binomial(n+6,6)/7); # G. C. Greubel, Oct 18 2019

Rn:=6; [2^(n-1)/(Rn+1)*Binomial(n+Rn,Rn)*(n+(Rn+1)*2) : n in [0..22]];

R:=PowerSeriesRing(Integers(), 23); Coefficients(R!( (1-x)/(1-2*x)^8 )); // Marius A. Burtea, Oct 17 2019

seq(2^(n-1)*(n+14)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Oct 18 2019

CoefficientList[Series[(1-x)/(1-2*x)^8, {x,0,30}], x] (* or *) Table[2^(n-1)*Binomial[n+6,6]*(n+14)/7, {n,0,30}] (* G. C. Greubel, Jan 03 2018 *)

for(n=0,30, print1(2^(n-1)*binomial(n+6,6)*(n+14)/7, ", ")) \\ G. C. Greubel, Jan 03 2018

a(n) = 2^(n-1)*binomial(n+6, 6)*(n+14)/7 = -A053120(n+14, n), n >= 0. [See a comment in A053120 on subdiagonal sequences. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^8. - Colin Barker, May 31 2013
E.g.f.: (1/315)*exp(2*x)*(315 + 4095*x + 11340*x^2 + 11550*x^3 + 5250*x^4 + 1134*x^5 + 112*x^6 + 4*x^7). - Stefano Spezia, Oct 17 2019

Name made more specific by Wolfdieter Lang, Nov 25 2019

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 0, 8, 0, 8, 0, 1, 16, 0, 20, 0, 5, 0, 32, 0, 48, 0, 18, 0, 1, 64, 0, 112, 0, 56, 0, 7, 0, 128, 0, 256, 0, 160, 0, 32, 0, 1, 256, 0, 576, 0, 432, 0, 120, 0, 9, 0, 512, 0, 1280, 0, 1120, 0, 400, 0, 50, 0, 1, 1024, 0, 2816, 0, 2816, 0, 1232, 0, 220

Offset: 0

Paul Barry, Mar 15 2003

Unsigned Chebyshev numbers of the first kind. Columns include A011782, A001792, A001793, A001794, A006974.

For the Riordan coefficient triangle for Chebyshev's T-polynomials (decreasing odd or even powers of x) see A039991. - Wolfdieter Lang, Aug 06 2014

			Triangle rows are {1}, {1,0}, {2,0,1}, {4,0,3,0}, {8,0,8,0,1},.... [Corrected by _Philippe Deléham_, Dec 27 2007]
See the unsigned example under A039991. - _Wolfdieter Lang_, Aug 06 2014

Cf. A008310, A039991 (signed).

T(n,k) = [x^k] a(n,x), k = 0, 1, ..., n, with polynomial a(n,x) defined by the recurrence given as name. Its Binet-de Moivre form is a(n, x) = ((1+sqrt(x^2+1))^n + (1-sqrt(x^2+1))^n)/2.

O.g.f. for row polynomials a(n,x): (1-z)/(1 - 2*z - (x*z)^2). Compare with A039991.

Edited. Name and formula clarified. G.f. of row polynomial, and crossref. A039991 added. - Wolfdieter Lang, Aug 06 2014

A304635 Triangle T(n,j) read by rows: the number of j-faces in the hypersimplicial decomposition of the unit cube of n dimensions.

Original entry on oeis.org

1, 5, 2, 18, 14, 3, 56, 64, 27, 4, 160, 240, 150, 44, 5, 432, 800, 660, 288, 65, 6, 1120, 2464, 2520, 1456, 490, 90, 7, 2816, 7168, 8736, 6272, 2800, 768, 119, 8, 6912, 19968, 28224, 24192, 13440, 4896, 1134, 152, 9, 16640, 53760, 86400, 86016, 57120, 25920, 7980, 1600, 189, 10

Offset: 1

R. J. Mathar, May 15 2018

			The triangle starts in row n>= for 1<=j<=n as:
  1,
5,2,
18,14,3,
56,64,27,4,
160,240,150,44,5,
432,800,660,288,65,6,
1120,2464,2520,1456,490,90,7,
2816,7168,8736,6272,2800,768,119,8,
6912,19968,28224,24192,13440,4896,1134,152,9,
16640,53760,86400,86016,57120,25920,7980,1600,189,10,

T. Hibi, N. Li, H. Ohsugi, The Face Vector of a Half-Open Hypersimplex, J. Int. Seq. 18 (2015) 15.6.6

Cf. A001793 (column j=1), A001794 (half of column j=2), A006974 (3rd of column j=3), A014106 (subdiagonal).

A304635 := proc(n,j)
        j*2^(n-j-1)*(n+j+2)/(n+1)*binomial(n+1,j+1) ;
end proc:

T(n,j) = j*2^(n-j-1)*(n+j+2)*binomial(n+,j+1)/(n+1).

cp's OEIS Frontend

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

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A209404 Negated coefficients of Chebyshev T polynomials: a(n) = -A053120(n+14, n), n >= 0.

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A081265 Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

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Author

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Comments

Examples

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Formula

Extensions

A304635 Triangle T(n,j) read by rows: the number of j-faces in the hypersimplicial decomposition of the unit cube of n dimensions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.