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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.

%I A122765 #23 Jan 26 2025 03:51:10
%S A122765 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,
%T A122765 -30,-6,7,4,-20,-30,60,30,-42,-7,8,5,20,-60,-60,105,42,-56,-8,9,-5,30,
%U A122765 60,-140,-105,168,56,-72,-9,10
%N 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.
%C A122765 Based on the coefficients of derivatives of the polynomials in A130777.
%H A122765 G. C. Greubel, <a href="/A122765/b122765.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A122765 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%F A122765 From _G. C. Greubel_, Dec 30 2022: (Start)
%F A122765 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))).
%F A122765 T(n, k) = k*(-1)^binomial(n-k+1, 2)*binomial(floor((n+k)/2), k).
%F A122765 T(n, n) = n.
%F A122765 T(n, n-1) = -(n-1).
%F A122765 T(n, n-2) = -2*A000217(n-2).
%F A122765 T(n, n-3) = 2*A000217(n-3).
%F A122765 T(n, 1) = (-1)^binomial(n, 2)*floor((n+1)/2).
%F A122765 T(n, 2) = 2*(-1)^binomial(n-1, 2)*binomial(floor((n+2)/2), 2).
%F A122765 Sum_{k=1..n} T(n, k) = A076118(n).
%F A122765 Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^(n-1)*A165202(n).
%F A122765 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = [n=1] - [n=2].
%F A122765 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)
%e A122765 Triangle begins as:
%e A122765    1;
%e A122765   -1,   2;
%e A122765   -2,  -2,   3;
%e A122765    2,  -6,  -3,   4;
%e A122765    3,   6, -12,  -4,   5;
%e A122765   -3,  12,  12, -20,  -5,   6;
%e A122765   -4, -12,  30,  20, -30,  -6,   7;
%e A122765    4, -20, -30,  60,  30, -42,  -7,   8;
%e A122765    5,  20, -60, -60, 105,  42, -56,  -8,  9;
%t A122765 (* First program *)
%t A122765 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
%t A122765 (* Second program *)
%t A122765 T[n_, k_]:= k*(-1)^Binomial[n-k+1,2]*Binomial[Floor[(n+k)/2], k];
%t A122765 Table[T[n, k], {n,14}, {k,n}]//Flatten (* _G. C. Greubel_, Dec 30 2022 *)
%o A122765 (PARI) tpol(n) = if (n<=0, 1, if (n==1, x-1, x*tpol(n-1) -tpol(n-2)));
%o A122765 lista(nn) = {for(n=0, nn, pol = deriv(tpol(n)); for (k=0, poldegree(pol), print1(polcoeff(pol, k), ", ");););} \\ _Michel Marcus_, Feb 07 2014
%o A122765 (Magma)
%o A122765 A122765:= func< n,k | k*(-1)^Binomial(n-k+1, 2)*Binomial(Floor((n+k)/2), k) >;
%o A122765 [A122765(n,k): k in [1..n], n in [1..14]]; // _G. C. Greubel_, Dec 30 2022
%o A122765 (SageMath)
%o A122765 def A122765(n, k): return k*(-1)^binomial(n-k+1, 2)*binomial(((n+k)//2), k)
%o A122765 flatten( [[A122765(n,k) for k in range(1,n+1)] for n in range(1,15)] ) # _G. C. Greubel_, Dec 30 2022
%Y A122765 Cf. A000217, A026741, A046854, A066170, A076118, A122766, A130777, A165202.
%K A122765 sign,tabl
%O A122765 1,3
%A A122765 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 22 2006
%E A122765 Name corrected and more terms from _Michel Marcus_, Feb 07 2014