login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A154649 A triangular sequence of coefficients: p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m - 1)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m + 3)^n*x^m, {m, 0, Infinity}])/2. 0
1, 1, 1, 5, -2, 5, 13, 11, 11, 13, 41, 108, 86, 108, 41, 121, 837, 962, 962, 837, 121, 365, 5258, 12163, 10508, 12163, 5258, 365, 1093, 30319, 130965, 160183, 160183, 130965, 30319, 1093, 3281, 165784, 1245980, 2503208, 2485414, 2503208, 1245980, 165784 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are:A000165

{1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200,...}.

The row sums are equivalent to the MacMahon numbers rows sums.

This results from a modular form bilinear approach summed:

f1(x)=(2*x+3)/(-x); f2(x)=(2*x-1)/(-x).

LINKS

Table of n, a(n) for n=0..43.

FORMULA

p(x,n)=((-1)^(-1 + n)* (-1 + x)(1 - n) *((-1)^n+2*n*LerchPhi[x, -n, 1/2])+

(-1)^(-1 + n)* 2^n* (-1 + x)(1 - n) LerchPhi[x, -n, 3/2])/2;

p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m - 1)^n*x^m, {m, 0, Infinity}] +

(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m + 3)^n*x^m, {m, 0, Infinity}])/2;

t(n,m)=coefficients(p(x,n))

EXAMPLE

{1},

{1, 1},

{5, -2, 5},

{13, 11, 11, 13},

{41, 108, 86, 108, 41},

{121, 837, 962, 962, 837, 121},

{365, 5258, 12163, 10508, 12163, 5258, 365},

{1093, 30319, 130965, 160183, 160183, 130965, 30319, 1093},

{3281, 165784, 1245980, 2503208, 2485414, 2503208, 1245980, 165784, 3281},

{9841, 878153, 10863860, 35584772, 45560654, 45560654, 35584772, 10863860, 878153, 9841},

{29525, 4558038, 89180081, 458019464, 852697082, 906922820, 852697082, 458019464, 89180081, 4558038, 29525}

MATHEMATICA

Clear[p]; p[x_, n_] = ((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m - 1)^n*x^m, {m, 0, Infinity}]

+ (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2*m + 3)^n*x^m, {m, 0, Infinity}])/2;

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%]

CROSSREFS

A000165

Sequence in context: A201530 A085997 A071546 * A100040 A197271 A248260

Adjacent sequences:  A154646 A154647 A154648 * A154650 A154651 A154652

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Jan 13 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 18 06:09 EST 2014. Contains 252079 sequences.