A137276 -id:A137276 - cp's OEIS frontend

A136255 Triangle T(n,k) read by rows: T(n,k) = (k+1) * A137276(n,k+1).

1, 0, 2, 1, 0, 3, 0, 0, 0, 4, -3, 0, -3, 0, 5, 0, -6, 0, -8, 0, 6, 5, 0, -6, 0, -15, 0, 7, 0, 16, 0, 0, 0, -24, 0, 8, -7, 0, 30, 0, 15, 0, -35, 0, 9, 0, -30, 0, 40, 0, 42, 0, -48, 0, 10, 9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11

Offset: 1

Roger L. Bagula, Mar 17 2008

Row sums are 1, 2, 4, 4, -1, -8, -9, 0, 12, 14, 1, ... with g.f. x*(1+3*x^2) / (x^2-x+1)^2.

			Triangle starts:
{1},
{0, 2},
{1, 0, 3},
{0, 0, 0, 4},
{-3, 0, -3, 0, 5},
{0, -6, 0, -8, 0, 6},
{5, 0, -6, 0, -15, 0, 7},
{0, 16, 0, 0, 0, -24, 0, 8},
{-7, 0, 30, 0, 15, 0, -35, 0, 9},
{0, -30, 0, 40, 0,42, 0, -48, 0, 10},
{9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11},
...

Cf. A138034, A135929, A135936, A137276, A137277, A137289.

B := proc(n,x) if n = 0 then 1; else add( (-1)^j*binomial(n-j,j)*(n-4*j)/(n-j)*x^(n-2*j),j=0..n/2) ; fi; end:
A136255 := proc(n,k) diff( B(n,x),x) ; coeftayl(%,x=0,k) ; end: seq( seq(A136255(n,k),k=0..n-1),n=1..15) ;

B[x, 0] = 1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 3] = x + x^3; B[x, 4] = -2 + x^4; B[x_, n_] := B[x, n] = x*B[x, n-1] - B[x, n-2]; P[x_, n_] := D[B[x, n + 1], x]; Flatten @ Table[CoefficientList[P[x, n], x], {n, 0, 10}]

T(n,k) = (k+1) * A137276(n,k+1) .

Edited by the Associate Editors of the OEIS, Aug 27 2009

Edited by and new name from Joerg Arndt, May 15 2016

A136256 Triangle of the coefficients [x^k] of the linear form (x*B_{n-1}(x)-(d/dx) B_n(x)) of the polynomials defined in A137276, 0<=k<=n.

Original entry on oeis.org

0, -1, 1, 0, -2, 1, -1, 2, -3, 1, 0, 0, 1, -4, 1, 3, -2, 3, 0, -5, 1, 0, 6, -3, 8, -1, -6, 1, -5, 2, 6, -3, 15, -2, -7, 1, 0, -16, 5, 0, -2, 24, -3, -8, 1, 7, -2, -30, 8, -15, 0, 35, -4, -9, 1, 0, 30, -7, -40, 10, -42, 3, 48, -5, -10, 1, -9, 2, 75, -15, -35, 10, -84, 7, 63, -6, -11, 1

Offset: 0

Roger L. Bagula, Mar 18 2008

Row sums are: 0, 0, -1, -1, -2, 0, 5, 7, 1, -9, -12, -2, 13, 17, 3, -17, -22, -4, 21, 27, 5

			0;
-1, 1;
0, -2, 1;
-1, 2, -3, 1;
0, 0, 1, -4, 1;
3, -2, 3, 0, -5, 1;
0, 6, -3, 8, -1, -6, 1;
-5, 2, 6, -3, 15, -2, -7, 1;
0, -16, 5, 0, -2, 24, -3, -8, 1;
7, -2, -30, 8, -15, 0, 35, -4, -9, 1;
0, 30, -7, -40, 10, -42, 3, 48, -5, -10, 1;
-9, 2, 75, -15, -35, 10, -84, 7, 63, -6, -11, 1;

Harry Hochstadt, Defined differential recursion, The Functions of Mathematical Physics, Dover (New York) (1986), page 49.

Cf. A138034.

B := proc(n,x) option remember; if n < 0 then 0; elif n = 0 then 1; elif n = 1 then x; elif n = 2 then x^2+2 else x*procname(n-1,x)-procname(n-2,x) ; expand(%) ; end if; end proc:
BB := proc(n,x) x*B(n-1,x)-diff(B(n,x),x) ; expand(%) ; end proc:
A136256 := proc(n,k) coeftayl(BB(n,x),x=0,k) ; end proc:
seq(seq(A136256(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Sep 04 2011

T(n,k) = A137276(n-1,k-1) - k*A137276(n,k+1). - R. J. Mathar, Sep 05 2011

A135929 Triangle read by rows: row n gives coefficients of polynomial P_n(x)= U_{n}(x,1) + 3 * U_{n-2}(x,1) where U is the Chebyshev polynomial of the second kind, in order of decreasing exponents.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 09 2008

Take a(0)=-2 instead of 1. The recurrence begins immediately (at the third instead of the fourth polynomial). Companion: A192011(n). - Paul Curtz, Sep 20 2011

			The coefficients and polynomials are
  1;                                 1
  1, 0;                              x
  1, 0,  2;                          x^2 + 2
  1, 0,  1, 0;                       x^3 +   x
  1, 0,  0, 0, -2;                   x^4 - 2
  1, 0, -1, 0, -3, 0;                x^5 -   x^3 - 3*x
  1, 0, -2, 0, -3, 0,  2;            x^6 - 2*x^4 - 3*x^2 + 2
  1, 0, -3, 0, -2, 0,  5, 0;         x^7 - 3*x^5 - 2*x^3 + 5*x
  1, 0, -4, 0,  0, 0,  8, 0, -2;     x^8 - 4*x^6 + 8*x^2 - 2
  1, 0, -5, 0,  3, 0, 10, 0, -7, 0;  x^9 - 5*x^7 + 3*x^5 + 10*x^3 - 7*x

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

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

Cf. A049310, A053119, A138034, A135936, A137276 (row-reversed), A192011, A194084, A219795.

A053119:= func< n,k | (1/2)*(-1)^Floor(3*k/2)*(1+(-1)^k)*Binomial(n - Floor(k/2), n-k) >;
A135929:= func< n,k | n eq 0 select 1 else A053119(n, k) + 3*A053119(n-2, k-2) >;
[A135929(n,k): k in [0..n], n in [0..16]]; // G. C. Greubel, Apr 24 2023

A135929 := proc(n, m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2), t=0, n), x=0, n-m) ; end proc: seq(seq(A135929(n,m), m=0..n),n=0..14) ; # R. J. Mathar, Nov 03 2009

p[0, ]= 1; p[1, x]:= x; p[2, x_]:= x^2+2; p[n_, x_]:= p[n, x] = x*p[n-1, x] - p[n-2, x]; row[n_]:= CoefficientList[p[n, x], x]; Table[row[n]//Reverse, {n, 0, 13}]//Flatten (* Jean-François Alcover, Nov 26 2012, after Paul Curtz's formula *)
(* Second program *)
p=1; q=2; t[, 0]=p; t[2, 2]=q; t[, ?OddQ]=0; t[n, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0]:= t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* Jean-François Alcover, Nov 27 2012, from recurrence *)

def A053119(n,k): return (-1)^(3*k/2)*((k+1)%2)*binomial(n-k/2, n-k)
def A135929(n,k): return 1 if (n==0) else A053119(n, k) + 3*A053119(n-2, k-2)
flatten([[A135929(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023

G.f.: (1+3*t^2)/(1-x*t+t^2).
P_n(x) = U_{n}(x,1) + 3 * U_{n-2}(x,1) for n>=2. - Max Alekseyev, Dec 04 2009
P_n(x) = S_{n}(x) + 3*S_{n-2}(x), with Chebyshev Polynomials S_n(x) defined in A049310 and A053119. - R. J. Mathar, Dec 07 2009
P_0(x)=1, P_1(x)=x, P_2(x)=x^2+2, and P_n(x)= x*P_{n-1}(x) - P_{n-2}(x) for n>=3. - Paul Curtz, Aug 14 2011
From G. C. Greubel, Apr 24 2023: (Start)
T(n, k) = A053119(n, k) + 3*A053119(n-2, k-2), with T(0,0) = 1.
Sum_{k=0..n} T(n, k) = A138034(n). (End)

Extended by R. J. Mathar, Nov 03 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.

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

A137277 Triangle of the coefficients [x^k] P_n(x) of the polynomials P_n(x) = 1/n * sum(j=0..floor(n/2), (-1)^j * binomial(n,j) * (n-4j) x^(n-2*j) ).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -6, 0, 0, 0, 1, 0, -6, 0, -1, 0, 1, 20, 0, -5, 0, -2, 0, 1, 0, 25, 0, -3, 0, -3, 0, 1, -70, 0, 28, 0, 0, 0, -4, 0, 1, 0, -98, 0, 28, 0, 4, 0, -5, 0, 1, 252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1, 0, 378, 0, -150, 0, 15, 0, 15, 0, -7, 0, 1, -924, 0, 528, 0, -165, 0, 0, 0, 22, 0, -8, 0, 1, 0, -1452

Offset: 0

Roger L. Bagula, Mar 13 2008

The first four P_n(x) are the same as in A137276.

Row sums are 1, 1, 3, 2, -5, -6, 14, 20, -45, -70, 154, a signed variant of A047074.

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-6, 0, 0, 0, 1}, = -6+x^4
{0, -6, 0, -1, 0, 1},
{20, 0, -5, 0, -2, 0, 1},
{0, 25, 0, -3,0, -3, 0, 1},
{-70, 0, 28, 0, 0, 0, -4, 0, 1},
{0, -98, 0, 28, 0,4, 0, -5, 0, 1},
{252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1}

Cf. A138034.

A137277 := proc(n,k) if n = 0 then 1; else add( (-1)^j*binomial(n,j)*(n-4*j)*x^(n-2*j),j=0..n/2)/n ; coeftayl(%,x=0,k) ; fi; end:
seq( seq(A137277(n,k),k=0..n),n=0..15) ;

B[x_, n_] = If[n > 0, Sum[(-1)^p*Binomial[n,p]*(n - 4*p)*x^(n - 2*p)/ n, {p, 0, Floor[n/2]}], 1]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a]

P(0,n)=1. P_n(x) = 1/n*sum(j=0..floor(n/2), (-1)^j*binomial(n,j)*(n-4*j)*x^(n-2*j)).

Edited by the Associate Editors of the OEIS, Aug 27 2009

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

A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.

Original entry on oeis.org

1, -2, 0, 10, -35, 84, -168, 300, -495, 770, -1144, 1638, -2275, 3080, -4080, 5304, -6783, 8550, -10640, 13090, -15939, 19228, -23000, 27300, -32175, 37674, -43848, 50750, -58435, 66960, -76384, 86768, -98175, 110670, -124320, 139194, -155363, 172900

Offset: 2

Jamel Ghanouchi, Nov 02 2009

The coefficient of [x^4] of the Polynomial B_{2n}(x) defined in A137276.

Essentially the same as A052472.

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5,-1).

Cf. A052472, A137276, A135929, A138034.

List([2..40], n-> (-1)^(n+1)*(n-4)*Binomial(n+1,3)/2); # G. C. Greubel, May 19 2019

[(-1)^(n+1)*n*(n-1)*(n-4)*(n+1)/12: n in [2..40]]; // Vincenzo Librandi, Jun 13 2016

Table[(-1)^(n+1)*(n+1)*n*(n-1)*(n-4)/12, {n, 2, 40}] (* G. C. Greubel, Jun 12 2016 *)
LinearRecurrence[{-5, -10, -10, -5, -1}, {1, -2, 0, 10, -35}, 40] (* Vincenzo Librandi, Jun 13 2016 *)

vector(40, n, n++; (-1)^(n+1)*(n-4)*binomial(n+1,3)/2) \\ G. C. Greubel, May 19 2019

[(-1)^(n+1)*(n-4)*binomial(n+1,3)/2 for n in (2..40)] # G. C. Greubel, May 19 2019

a(n) = -5*a(n-1) -10*a(n-2) -10*a(n-3) -5*a(n-4) -a(n-5).
G.f.: x^2*(1+3*x)/(1+x)^5.
E.g.f.: x^2*(6 + 2*x - x^2)*exp(-x)/12. - G. C. Greubel, May 19 2019

cp's OEIS Frontend

A136255 Triangle T(n,k) read by rows: T(n,k) = (k+1) * A137276(n,k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A136256 Triangle of the coefficients [x^k] of the linear form (x*B_{n-1}(x)-(d/dx) B_n(x)) of the polynomials defined in A137276, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Formula

A135929 Triangle read by rows: row n gives coefficients of polynomial P_n(x)= U_{n}(x,1) + 3 * U_{n-2}(x,1) where U is the Chebyshev polynomial of the second kind, in order of decreasing exponents.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Keywords

Comments

Links

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

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

A137277 Triangle of the coefficients [x^k] P_n(x) of the polynomials P_n(x) = 1/n * sum(j=0..floor(n/2), (-1)^j * binomial(n,j) * (n-4j) x^(n-2*j) ).

A167387 a(n) = (-1)^(n+1) * n(n-1)(n-4)*(n+1)/12.