A135929 -id:A135929 - cp's OEIS frontend

A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

2, 2, 2, 2, 3, 3, 5, 7, 10, 11, 16, 23, 33, 44, 58, 81, 114, 158, 212, 293, 407, 565, 777, 1064, 1471, 2036, 2813, 3863, 5334, 7370, 10183, 14046, 19356, 26726, 36909, 50955, 70251, 96977, 133886, 184841, 255092

Offset: 0

Paul Curtz, Nov 28 2012

nonn

The companion to A135929(n) is the triangle
2;
2, 0;
2, 0,  1;
2, 0, -1, 0;
2, 0, -3, 0, -1;
2, 0, -5, 0,  0, 0;
2, 0, -7, 0,  3, 0, 1;
2, 0, -9, 0,  8, 0, 1, 0;
(A192011(n) beginning with 2 instead of -1).
Consider a(1),a(5),a(10),a(14), that is, a(A193910(n) -1).
a(1)+a(4)-a(5) = 2, a(5)+a(8)-a(9) = 2, a(10)+a(13)-a(14) = 2, a(14)+a(17)-a(18) = 4, a(19)+a(22)-a(23) = 6, a(23)+a(26)-a(27) = 14, yields 2,2,2,4,6,14,24,60,... = 2*A047749(n) or 2, followed with A116637(n+1).

			a(0)=2, a(1)=2, a(2)=2+0, a(3)=2+0, a(4)=2+0+1, a(5)=2+0+1.

A219795 := proc(n)
    if n=0 then
        2;
    else
        add(abs(A192011(n-k,k)),k=0..floor(n/2)) ;
    end if;
end proc: # R. J. Mathar, Jan 06 2013

a(n) = sum abs ( [k=0..floor(n/2)] A192011(n-k,k) ), a(0)=2.

a(24)-a(40) from Jean-Francois Alcover, Nov 28 2012

A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

Original entry on oeis.org

0, 3, 0, 3, 0, 3, 3, 0, 0, 0, 3, 0, -3, 0, -3, 3, 0, -6, 0, -3, 0, 3, 0, -9, 0, 0, 0, 3, 3, 0, -12, 0, 6, 0, 6, 0, 3, 0, -15, 0, 15, 0, 6, 0, -3, 3, 0, -18, 0, 27, 0, 0, 0, -9, 0

Offset: 0

Paul Curtz, Aug 14 2011

0,
3, 0,
3, 0,  3,
3, 0,  0, 0,
3, 0, -3, 0, -3,
3, 0, -6, 0, -3, 0.
Multiples of 3.
Row sum (from the second) is period 6: 3*A057079(n),"from" A057083 (scaled Chebyshev U(n,x)).
If a(0)=-3, a(n)=3*A192174(n).

			a(0)=1-1=0, a(1)=1+2=3, a(2)=0+0=0.

BC(0,x)=0, BC(1,x)=3*x, BC(2,x)=3*x^2+3, BC(n,x)=x*BC(n-1,x) - BC(n-2,x), n > 2.

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -2, 0, 0, 0, 1, 0, -3, 0, -1, 0, 1, 2, 0, -3, 0, -2, 0, 1, 0, 5, 0, -2, 0, -3, 0, 1, -2, 0, 8, 0, 0, 0, -4, 0, 1, 0, -7, 0, 10, 0, 3, 0, -5, 0, 1, 2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1, 0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1, -2, 0, 24, 0, -35, 0, 0, 0, 18, 0, -8, 0, 1, 0, -11, 0, 49, 0, -42, 0, -12, 0

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 13 2008

Polynomial coefficients of P(n,x) in increasing powers, read by rows, where P(0,x)=1, P(1,x)=x, P(2,x)=2+x^2, P(3,x)=x+x^3, P(4,x)=-2+x^4, and  P(n,x) = x*P(n-1,x) - P(n-2,x) for n>=5.
The row-reversed version of A135929.
Row sums are repeating 1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1..., see A138034 and A119910.

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-2, 0, 0, 0, 1}, = -2+x^4
{0, -3, 0, -1, 0, 1}, = -3x-x^3+x^5
{2, 0, -3, 0, -2, 0, 1},
{0, 5, 0, -2, 0, -3, 0, 1},
{-2, 0, 8, 0, 0, 0, -4, 0, 1},
{0, -7, 0, 10, 0, 3, 0, -5, 0, 1},
{2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1},
{0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165

Cf. A123956, A137289.

A137276 := proc(n,k) local nmk,npk; if n = 0 then 1; elif (n-k) mod 2 <> 0 then 0; else nmk := (n-k)/2 ; npk := (n+k)/2 ; (-1)^nmk*(2*k-n)/npk*binomial(npk,nmk) ; fi; end:
seq( seq(A137276(n,k),k=0..n),n=0..13) ;

T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.
P(n,x) = x*P(n-1,x)-P(n-2,x), n>=5.
P(n,2*x) = -2*T(n,x)+4*x*U(n-1,x), where T(n,x) is A053120 and U(n,x) is A053117.

Fourth row inserted by the Associate Editors of the OEIS, Aug 27 2009

A138034 Expansion of (1+3*x^2)/(1-x+x^2).

Original entry on oeis.org

1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3

Offset: 0

Karem Boubaker (mmbb11112000(AT)yahoo.fr), Mar 01 2008; corrected Mar 03 2008

Essentially a duplicate of A119910: 1, followed by A119910. - Joerg Arndt, Nov 14 2014

Karem Boubaker and Lin Zhang, Fermat-linked relations for the Boubaker polynomial sequences via Riordan matrices analysis, arXiv preprint arXiv:1203.2082, 2012. - From _N. J. A. Sloane_, Sep 15 2012
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A135929, A135936, A137276.

CoefficientList[Series[(1 + 3*x^2)/(1 - x + x^2), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jan 15 2017 *)
LinearRecurrence[{1,-1},{1,1,3},120] (* Harvey P. Dale, Jun 14 2024 *)

a(n) = A119910(n), n>=1.

G.f.: (1+3*x^2)/(1-x+x^2). a(n)=a(n-1)-a(n-2), n>2.

A135936 Irregular triangle read by rows: row n gives coefficients of Boubaker polynomial B_n(x) in order of decreasing exponents (another version).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 0, -2, 1, -1, -3, 1, -2, -3, 2, 1, -3, -2, 5, 1, -4, 0, 8, -2, 1, -5, 3, 10, -7, 1, -6, 7, 10, -15, 2, 1, -7, 12, 7, -25, 9, 1, -8, 18, 0, -35, 24, -2, 1, -9, 25, -12, -42, 49, -11, 1, -10, 33, -30, -42, 84, -35, 2, 1, -11, 42, -55, -30, 126, -84, 13, 1, -12, 52, -88, 0, 168, -168, 48, -2, 1, -13, 63, -130, 55, 198, -294

Offset: 0

N. J. A. Sloane, Mar 09 2008

See A135929 and A138034 for further information.

			The Boubaker polynomials B_0(x), B_1(x), B_2(x), ... are:
  1
  x
  x^2    + 2
  x^3    + x
  x^4             - 2
  x^5    - x^3  - 3*x
  x^6  - 2*x^4  - 3*x^2    + 2
  x^7  - 3*x^5  - 2*x^3  + 5*x
  x^8  - 4*x^6           + 8*x^2    - 2
  x^9  - 5*x^7  + 3*x^5 + 10*x^3  - 7*x
  ...

R. J. Mathar, Mar 11 2008, Table of n, a(n) for n = 0..160

Cf. A138034.

A135936 := proc(n,m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2),t=0,n), x=0,m) ; end: for n from 0 to 25 do for m from n to 0 by -2 do printf("%d, ",A135936(n,m)) ; od; od; # R. J. Mathar, Mar 11 2008

T[n_, m_] := SeriesCoefficient[SeriesCoefficient[
   (1+3*t^2)/(1-x*t+t^2), {t, 0, n}], {x, 0, m}];
Table[T[n, m], {n, 0, 25}, {m, n, 0, -2}] // Flatten (* Jean-François Alcover, Mar 11 2023, after R. J. Mathar *)

Conjectures from Thomas Baruchel, Jun 03 2018: (Start)
T(n,m) = 4*A115139(n+1,m) - 3*A132460(n,m).
T(n,m) = (-1)^m * (binomial(n-m, m) - 3*binomial(n-m-1, m-1)). (End)

More terms from R. J. Mathar, Mar 11 2008

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

Original entry on oeis.org

2, 5, 8, 10, 10, 7, 0, -12, -30, -55, -88, -130, -182, -245, -320, -408, -510, -627, -760, -910, -1078, -1265, -1472, -1700, -1950, -2223, -2520, -2842, -3190, -3565, -3968, -4400, -4862, -5355, -5880, -6438, -7030, -7657, -8320, -9020, -9758, -10535, -11352

Offset: 6

Jamel Ghanouchi, Nov 06 2009

Essentially the same as A111396.

The coefficient of x^(n-6) of the Polynomial B_n(x) defined in A135929.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A135929, A137276.

LinearRecurrence[{4,-6,4,-1},{2,5,8,10},50] (* Harvey P. Dale, May 27 2012 *)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^6*(2 - 3*x)/(x - 1)^4.
a(n) = -A111396(n-12) for n > 11. - Bruno Berselli, Oct 02 2018

Minor edits by N. J. A. Sloane, Nov 09 2009

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

A167373 Expansion of (1+x)(3x+1)/(1+x+x^2).

Original entry on oeis.org

1, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1, -2, 3, -1

Offset: 0

Jamel Ghanouchi, Nov 02 2009

Bisection of A138034.

Also row 2n of A137276 or A135929.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 22.

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

Cf. A135929, A138034, A137276.

A167373 := proc(n)
    option remember;
    if n < 4 then
        op(n+1,[1,3,-1,-2]) ;
    else
        procname(n-3) ;
    end if;
end proc:
seq(A167373(n),n=0..20) ; # R. J. Mathar, Feb 06 2020

CoefficientList[Series[(1 + x)*(3*x + 1)/(1 + x + x^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-1,-1},{1,3,-1},120] (* Harvey P. Dale, Apr 05 2023 *)

G.f.: (1+x)*(3*x+1)/(1+x+x^2).
a(n) = a(n-3), n>4.
a(n) = - a(n-1) - a(n-2) for n>2.
a(n) = 4*sin(2*n*Pi/3)/sqrt(3)-2*cos(2*n*Pi/3) for n>0 with a(0)=1. - Wesley Ivan Hurt, Jun 12 2016

Edited by R. J. Mathar, Nov 03 2009
Further edited and extended by Simon Plouffe, Nov 23 2009
Recomputed by N. J. A. Sloane, Dec 20 2009

A167544 a(n) = (n-3)*(n-8)/2.

Original entry on oeis.org

-2, -3, -3, -2, 0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173

Offset: 4

Jamel Ghanouchi, Nov 06 2009

Essentially a duplicate of A055998.

G. C. Greubel, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A027379, A055998, A135929, A137276.

[(n-3)*(n-8)/2: n in [4..60]]; // G. C. Greubel, Jul 30 2022

Table[(n-3)*(n-8)/2, {n,4,60}] (* G. C. Greubel, Jun 15 2016 *)

a(n)=(n-3)*(n-8)/2 \\ Charles R Greathouse IV, Jun 17 2017

[(n-3)*(n-8)/2 for n in (4..60)] # G. C. Greubel, Jul 30 2022

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x^4*(-2 + 3*x)/(1-x)^3.
a(n) = A055998(n-8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n - 6 (with a(4)=-2). - Vincenzo Librandi, Dec 05 2010
a(n) = A027379(n-8) for n >= 9. - Georg Fischer, Oct 24 2018
E.g.f.: (1/2)*( (24 -10*x + x^2)*exp(x) - (24 + 14*x + 3*x^2) ). - G. C. Greubel, Jul 30 2022

Edited and extended by R. J. Mathar, Nov 12 2009

A137289 Triangle read by rows: T(n,k) = (-1)^(n-k)(C(k+n-1,n-k)-2C(k+n-1,n-k-1)) for n>=0 and 0<=k<=n.

Original entry on oeis.org

-1, 2, 1, -2, 0, 1, 2, -3, -2, 1, -2, 8, 0, -4, 1, 2, -15, 10, 7, -6, 1, -2, 24, -35, 0, 18, -8, 1, 2, -35, 84, -42, -30, 33, -10, 1, -2, 48, -168, 168, 0, -88, 52, -12, 1, 2, -63, 300, -462, 198, 143, -182, 75, -14, 1, -2, 80, -495, 1056, -858, 0, 455, -320, 102, -16, 1

Offset: 1

Roger L. Bagula, Mar 14 2008

Previous name was: "Expansion of certain polynomials; see formula."

			{-1},
{2, 1},
{-2, 0, 1},
{2, -3, -2, 1},
{-2, 8, 0, -4, 1},
{2, -15, 10, 7, -6, 1},
{-2, 24, -35, 0, 18, -8, 1},
{2, -35, 84, -42, -30, 33, -10, 1},
{-2, 48, -168, 168,0, -88, 52, -12, 1},
{2, -63, 300, -462, 198, 143, -182, 75, -14,1},
{-2, 80, -495, 1056, -858, 0, 455, -320, 102, -16, 1}

Cf. A135929, A138034.

T := (n,k) -> (-1)^(n-k)*(binomial(k+n-1,n-k)-2*binomial(k+n-1,n-k-1)):
seq(seq(T(n,k), k=0..n), n=0..10); # Peter Luschny, May 15 2016

B[x, 0] = -1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 4] = -2 + x^4; B[ x, 3] = x + x^3; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2]; a = Table[CoefficientList[B[x, n] /. x -> Sqrt[y], y], {n, 0, 20, 2}]; Flatten[a]

B(x, 0) = -1, B(x, 2) = x^2 + 2, B(x, 3) = x^3 + x, B(x, 4) = x^4 - 2, and B(x, n) = x*B(x, n - 1) - B(x, n - 2) for n>=2, expand B(sqrt(x), 2*n).

Edited by N. J. A. Sloane, Jan 05 2009
Edited by Joerg Arndt, Nov 15 2014
New name and changed a(1) to -1 by Peter Luschny, May 15 2016

A161718 Expansion of (1+3*x^2)/(1+x)^2.

Original entry on oeis.org

1, -2, 6, -10, 14, -18, 22, -26, 30, -34, 38, -42, 46, -50, 54, -58, 62, -66, 70, -74, 78, -82, 86, -90, 94, -98, 102, -106, 110, -114, 118, -122, 126, -130, 134, -138, 142, -146, 150, -154, 158, -162, 166, -170, 174, -178, 182, -186, 190, -194, 198, -202, 206, -210, 214, -218, 222, -226, 230, -234

Offset: 0

Pr Mosbah Amlouk (Mosbah.Amlouk(AT)fsb.rnu.tn), Jun 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A016825, A111284 (unsigned version), A135929, A137276.

CoefficientList[Series[(1+3*x^2)/(1+x)^2,{x,0,80}],x] (* or *) LinearRecurrence[ {-2,-1},{1,-2,6},80] (* Harvey P. Dale, Mar 19 2016 *)

x='x+O('x^99); Vec((1+3*x^2)/(1+x)^2) \\ Altug Alkan, Apr 17 2018

From R. J. Mathar, Aug 27 2009: (Start)
a(n) = -2*a(n-1)-a(n-2), n>3.
G.f.: (1+3*x^2)/(1+x)^2.
a(n) = 4*(-1)^n*n+2*(-1)^(n+1) = (-1)^n*A016825(n-1), n>0. (End)
E.g.f.: 3 - 2*exp(-x)*(1 + 2*x). - Stefano Spezia, Feb 02 2023

Spurious commas in sequence deleted by N. J. A. Sloane, Aug 02 2009
Offset corrected, extended by R. J. Mathar, Aug 27 2009
Edited by Joerg Arndt, Sep 04 2011

cp's OEIS Frontend

A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

Views

Author

Keywords

Comments

Examples

Programs

Maple

Formula

Extensions

A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

Views

Author

Keywords

Comments

Examples

Formula

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A138034 Expansion of (1+3*x^2)/(1-x+x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A135936 Irregular triangle read by rows: row n gives coefficients of Boubaker polynomial B_n(x) in order of decreasing exponents (another version).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A167541 a(n) = -(n - 4)*(n - 5)*(n - 12)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A167373 Expansion of (1+x)*(3*x+1)/(1+x+x^2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A167544 a(n) = (n-3)*(n-8)/2.

Views

Author

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

A167541 a(n) = -(n - 4)(n - 5)(n - 12)/6.

A167373 Expansion of (1+x)(3x+1)/(1+x+x^2).

A137289 Triangle read by rows: T(n,k) = (-1)^(n-k)(C(k+n-1,n-k)-2C(k+n-1,n-k-1)) for n>=0 and 0<=k<=n.