A157047 A triangle of infinite sum coefficients with: Limit[Log[1-x],x->0]=-x: p(x,y)=1+n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}]; such that Log[1-x]->-x.

2, 1, 1, 1, -1, 2, 1, 3, -7, 6, 1, -12, 40, -46, 24, 1, 60, -260, 430, -326, 120, 1, -360, 1920, -4140, 4536, -2556, 720, 1, 2520, -15960, 42420, -60732, 49644, -22212, 5040, 1, -20160, 147840, -467040, 825216, -883008, 574848, -212976, 40320, 1, 181440

Offset

0,1

Comments

Row sums are:1+n!;

{2, 2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881,...}.

Links

Table of n, a(n) for n=0..46.

Formula

Limit[Log[1-x],x->0]=-x:

p(x,y)=1+n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}];

such that Log[1-x]->-x.

Example

{2},
{1, 1},
{1, -1, 2},
{1, 3, -7, 6},
{1, -12, 40, -46, 24},
{1, 60, -260, 430, -326, 120},
{1, -360, 1920, -4140, 4536, -2556, 720},
{1, 2520, -15960, 42420, -60732, 49644, -22212, 5040},
{1, -20160, 147840, -467040, 825216, -883008, 574848, -212976, 40320},
{1, 181440, -1512000, 5533920, -11630304, 15374016, -13120704, 7090416, -2239344, 362880},
{1, -1814400, 16934400, -70459200, 171642240, -270043200, 284947200, -202111200, 93297600, -25659360, 3628800}

Mathematica

Clear[p, x, n, m];
p[x_, n_] = n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}]
Table[ExpandAll[1 + p[x, n] /. Log[1 - x] -> -x], {n, 0, 10}]
Table[CoefficientList[ExpandAll[1 + p[x, n] /. Log[1 - x] -> - x], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A156281 A002217 A344173 * A059342 A357481 A062831

Adjacent sequences: A157044 A157045 A157046 * A157048 A157049 A157050

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Feb 22 2009

Status

approved