A127946 Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.

1, -6, -108, 5832, 944784, -459165024, -669462604992, 2928229434235008, 38424226636031774976, -1512608105754026853705216, -178635992073339063368878599168, 63289660175631590117213474413627392, 67269440586795655766964092111705109663744

Offset: 0

Author

Paul Barry, Feb 08 2007

Keywords

• easy
• sign

Comments

Hankel transform of A098333. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-3)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).

Let T_n denote the n X n matrix with T_n(i,j) = 3^min(i,j); then a(n) = ((-1)^floor((n+1)/2))*det(T_(n+1))/3. - Lechoslaw Ratajczak, May 16 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..63

Crossrefs

a(n) = A083667(n+1)/2.

Programs

Magma

[2^n*(-3)^Binomial(n+1,2): n in [0..30]]; // G. C. Greubel, May 03 2018

Mathematica

A127946[0] = 1; A127946[n_] := {1, -1, -1, 1}[[Mod[n, 4] + 1]] * 2^n * 3^(n(n + 1)/2); Table[A127946[n], {n, 0, 12}] (* Jean-François Alcover, Oct 04 2016 *)
Table[2^n*(-3)^Binomial[n+1,2], {n,0,30}] (* G. C. Greubel, May 03 2018 *)

PARI

a(n)=if((n-1)%4<2,-1,1)*2^n*3^(n*(n+1)/2) \\ Charles R Greathouse IV, Oct 04 2016

Formula

a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*6^n*3^C(n,2) = 2^n*(-3)^C(n+1,2).