A137862 Triangular sequence of coefficients of the expansion of a degenerate partition of Chebyshev U(x,n);A053117 and Hermite H(x,n);A060821 functions: 1) f(x,t)=1/(1-2*x*t+t^2); 2) g(x,t)=Exp[2*x*t-t^2]; to give: p(x,t)=Exp[2*x*t-t^2]/(1-2*x*t+t^2).

1, 0, 4, -4, 0, 20, 0, -60, 0, 128, 60, 0, -768, 0, 1040, 0, 1920, 0, -10400, 0, 10432, -1920, 0, 46800, 0, -156480, 0, 125248, 0, -109200, 0, 1095360, 0, -2630208, 0, 1753600, 109200, 0, -4381440, 0, 26302080, 0, -49100800, 0, 28057856, 0, 9858240, 0, -157812480, 0, 662860800, 0, -1010082816, 0

Offset: 1

Author

Roger L. Bagula, Apr 29 2008

Keywords

• tabl
• uned
• sign

Comments

Row sums are:

{1, 4, 16, 68, 332, 1952, 13648, 109552, 986896, 9865664, 108500864};

Examples

			{1},
{0, 4},
{-4, 0, 20},
{0, -60, 0, 128},
{60, 0, -768, 0,1040},
{0, 1920, 0, -10400, 0, 10432},
{-1920, 0, 46800, 0, -156480, 0, 125248},
{0, -109200, 0, 1095360, 0, -2630208, 0, 1753600},
{109200, 0, -4381440, 0, 26302080, 0, -49100800, 0, 28057856},
{0, 9858240, 0, -157812480, 0, 662860800, 0, -1010082816, 0, 505041920}, {-9858240, 0, 591796800, 0, -5523840000, 0, 17676449280, 0, -22726886400, 0, 10100839424}
		

Crossrefs

Cf. A060821, A053117.

Programs

Mathematica

Clear[p, b, a]; p[t_] = FullSimplify[(1/(1 - 2*x*t + t^2))*Exp[2*x*t - t^2]]; Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Formula

p(x,t)=Exp[2*x*t-t^2]/(1-2*x*t+t^2)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}); out_n,m=n!*Coefficients(P(x,n)).