A138034 -id:A138034 - cp's OEIS frontend

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.

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

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

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

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

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

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

A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

Original entry on oeis.org

1, 3, 11, 30, 79, 207, 542, 1419, 3715, 9726, 25463, 66663, 174526, 456915, 1196219, 3131742, 8199007, 21465279, 56196830, 147125211, 385178803, 1008411198, 2640054791, 6911753175, 18095204734, 47373861027, 124026378347, 324705274014, 850089443695

Offset: 0

Jamel Ghanouchi, Nov 02 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1).

Cf. A135929, A138034.

I:=[1,3,11]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 26 2014

Join[{1},LinearRecurrence[{3,-1},{3,11},30]] (* Harvey P. Dale, Jun 25 2014 *)
CoefficientList[Series[(3 x^2 + 1)/(1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2014 *)
Table[3LucasL[2n+1]-Fibonacci[2n], {n,0,20}] (* Rigoberto Florez, Dec 24 2018 *)

a(n) = (-1)^n*A098150(n-1), n>0.
G.f.: (3*x^2+1)/(1-3*x+x^2).
a(n) = 3*L(2n+1)-F(2n), where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - Rigoberto Florez, Dec 24 2018

Edited by R. J. Mathar, Nov 03 2009

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

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.

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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.

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

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

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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-4*j) * x^(n-2*j) ).

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

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A167375 a(n)=3*a(n-1)-a(n-2) with a(0)=1, a(1)=3, a(2)=11.

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.

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) ).

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.

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