A122765 Triangle read by rows: Let p(k, x) = x*p(k-1, x) - p(k-2, x). Then T(k,x) = dp(k,x)/dx.

1, -1, 2, -2, -2, 3, 2, -6, -3, 4, 3, 6, -12, -4, 5, -3, 12, 12, -20, -5, 6, -4, -12, 30, 20, -30, -6, 7, 4, -20, -30, 60, 30, -42, -7, 8, 5, 20, -60, -60, 105, 42, -56, -8, 9, -5, 30, 60, -140, -105, 168, 56, -72, -9, 10

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 22 2006

Keywords

• sign
• tabl

Comments

Based on the coefficients of derivatives of the polynomials in A130777.

Examples

			Triangle begins as:
   1;
  -1,   2;
  -2,  -2,   3;
   2,  -6,  -3,   4;
   3,   6, -12,  -4,   5;
  -3,  12,  12, -20,  -5,   6;
  -4, -12,  30,  20, -30,  -6,   7;
   4, -20, -30,  60,  30, -42,  -7,   8;
   5,  20, -60, -60, 105,  42, -56,  -8,  9;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Crossrefs

Cf. A000217, A026741, A046854, A066170, A076118, A122766, A130777, A165202.

Programs

Magma

A122765:= func< n,k | k*(-1)^Binomial(n-k+1, 2)*Binomial(Floor((n+k)/2), k) >;
[A122765(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Dec 30 2022

Mathematica

(* First program *)
p[0,x]=1; p[1,x]=x-1; p[k_,x_]:= p[k, x]= x*p[k-1,x] -p[k-2,x]; a = Table[Expand[p[n, x]], {n, 0, 10}]; Table[CoefficientList[D[a[[n]], x], x], {n, 2, 10}]//Flatten
(* Second program *)
T[n_, k_]:= k*(-1)^Binomial[n-k+1,2]*Binomial[Floor[(n+k)/2], k];
Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, Dec 30 2022 *)

PARI

tpol(n) = if (n<=0, 1, if (n==1, x-1, x*tpol(n-1) -tpol(n-2)));
lista(nn) = {for(n=0, nn, pol = deriv(tpol(n)); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", ");););} \\ Michel Marcus, Feb 07 2014

SageMath

def A122765(n, k): return k*(-1)^binomial(n-k+1, 2)*binomial(((n+k)//2), k)
flatten( [[A122765(n,k) for k in range(1,n+1)] for n in range(1,15)] ) # G. C. Greubel, Dec 30 2022

Formula

From G. C. Greubel, Dec 30 2022: (Start)

T(n, k) = coefficient [x^k]( p(n, x) ), where p(n,x) = (2/(x^2-4))*((n+1)*chebyshev_T(n+1,x/2) -n*chebyshev_T(n,x/2) - (x/2)*(chebyshev_U(n,x/2) - chebyshev_U(n-1,x/2))).

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

T(n, n) = n.

T(n, n-1) = -(n-1).

T(n, n-2) = -2*A000217(n-2).

T(n, n-3) = 2*A000217(n-3).

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

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

Sum_{k=1..n} T(n, k) = A076118(n).

Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^(n-1)*A165202(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = [n=1] - [n=2].

Sum_{k=1..floor((n+1)/2)} (-1)^k*T(n-k+1, k) = (-1)^binomial(n+1, 2)*b(n), where b(n) = 4^floor(n/4)*A026741(n/2) if n is even and b(n) = 4^floor((n-1)/4)*A026741((n-1)/4) if n is odd. (End)

Extensions

Name corrected and more terms from Michel Marcus, Feb 07 2014