A136526 - OEIS

%I #11 Sep 23 2022 03:09:55

%S 1,0,1,-6,0,6,0,-42,0,36,36,0,-288,0,216,0,468,0,-1944,0,1296,-216,0,

%T 4536,0,-12960,0,7776,0,-4104,0,38880,0,-85536,0,46656,1296,0,-51840,

%U 0,311040,0,-559872,0,279936,0,32400,0,-544320,0,2379456,0,-3639168,0,1679616

%N Coefficients polynomials B(x, n) = ((1 + a + b)*x - c)*B(x, n-1) - a*b*B(x, n-2) with a = 3, b = 2, and c = 0.

%D Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

%H G. C. Greubel, <a href="/A136526/b136526.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.

%F From _G. C. Greubel_, Sep 22 2022: (Start)

%F T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).

%F T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.

%F T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).

%F T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.

%F T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.

%F T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.

%F T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.

%F T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.

%F T(n, n) = 6^(n-1) + (5/6)*[n=0].

%F T(n, n-2) = -6*A081106(n-2), n >= 2.

%F Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.

%F Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].

%F G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

%e Triangle begins as:

%e      1;

%e      0,     1;

%e     -6,     0,      6;

%e      0,   -42,      0,    36;

%e     36,     0,   -288,     0,    216;

%e      0,   468,      0, -1944,      0,   1296;

%e   -216,     0,   4536,     0, -12960,      0,    7776;

%e      0, -4104,      0, 38880,      0, -85536,       0, 46656;

%e   1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

%t (* First program *)

%t a= (b+1)/(b-1); c=0; b=2;

%t B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];

%t Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten

%t (* Second program *)

%t B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);

%t Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* _G. C. Greubel_, Sep 22 2022 *)

%o (Magma)

%o f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;

%o A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;

%o [A136526(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 22 2022

%o (SageMath)

%o def f(n,k):

%o     if (k==0): return (-1)^(n//2)

%o     else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)

%o def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)

%o flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 22 2022

%Y Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

%K tabl,sign

%O 0,4

%A _Roger L. Bagula_, Mar 23 2008

%E Edited by _G. C. Greubel_, Sep 22 2022