A137455 A triangular sequence of coefficients from a three level exponential expansion function: f(x,t)=Log(1 + t)*(1 - t)*Exp(x*(t - t^2)).

0, 1, -3, 2, 5, -15, 3, -14, 56, -42, 4, 54, -170, 290, -90, 5, -264, 744, -1350, 1000, -165, 6, 1560, -4116, 6174, -7210, 2695, -273, 7, -10800, 27264, -37296, 41664, -28420, 6160, -420, 8, 85680, -209520, 270864, -260064, 223524, -89964, 12516, -612, 9, -766080, 1828800, -2274480, 2021760, -1587600

Offset

1,3

Comments

Row sums are:

{0, 1, -1, -7, 4, 89, -29, -1163, -1840, 32433, -38897}

The idea is that the three exponential levels are:

1) Log(1+t)

2) (1-t)

3) Exp(x*(t-t^2))

Links

Table of n, a(n) for n=1..51.

Formula

n! times Coefficients of the polynomial expansion: f(x,t)=Log(1 + t)*(1 - t)*Exp(x*(t - t^2))=Sum[(p(x,n)*t^n/n!,{n,0,Infinity}].

Example

{0},
{1},
{-3, 2},
{5, -15, 3},
{-14, 56, -42,4},
{54, -170, 290, -90, 5},
{-264, 744, -1350, 1000, -165, 6},
{1560, -4116, 6174, -7210, 2695, -273, 7},
{-10800, 27264, -37296, 41664, -28420, 6160, -420, 8},
{85680, -209520, 270864, -260064, 223524, -89964, 12516, -612, 9},
{-766080, 1828800, -2274480, 2021760, -1587600, 958608, -242340, 23280, -855, 10}

Mathematica

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

Crossrefs

Sequence in context: A265759 A057674 A092935 * A111273 A068553 A174909

Adjacent sequences: A137452 A137453 A137454 * A137456 A137457 A137458

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 18 2008

Status

approved