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

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A167883 Coefficients of a recursive polynomial:p(k,n)=If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))] ( correction with cubic term in the infinite sum) 0
1, 1, 2, 1, 10, 10, 1, 32, 90, 60, 1, 74, 594, 1040, 520, 1, 224, 2226, 6684, 7800, 3120, 1, 352, 12124, 95304, 227052, 215280, 71760, 1, 1058, 38484, 358656, 1252980, 2008152, 1506960, 430560, 1, 1348, 142264, 4028712, 32909556, 97352640 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums are:

{1, 3, 21, 183, 2229, 20055, 621873, 5596851, 374989401, 3374904603, 422613054909,...}

The remarkable thing about these polynomials is that there infinite sums are a symmetrical triangle.

Quadratic {1,18,1} type approximates the general Pascal recursion which is the first,

since the MacMahon not to be a rational inverse z Transform.

LINKS

Table of n, a(n) for n=0..41.

FORMULA

p(k,0)=1;

p(k,1)=1+2*k; p(k,n)=If[Mod[n, 2] == 0, (1 + 2*k)*p(k, n - 1) + n*Binomial[n + 1, n - 1]*k*(k + 1)*p(k, n - 2), (1 + 2*k)*(1 + 3*(p(k, n - 1) - 1))]

EXAMPLE

{1},

{1, 2},

{1, 10, 10},

{1, 32, 90, 60},

{1, 74, 594, 1040, 520},

{1, 224, 2226, 6684, 7800, 3120},

{1, 352, 12124, 95304, 227052, 215280, 71760},

{1, 1058, 38484, 358656, 1252980, 2008152, 1506960, 430560},

{1, 1348, 142264, 4028712, 32909556, 97352640, 132914880, 86112000, 21528000},

{1, 4046, 434880, 12939720, 122900940, 489515256, 982860480, 1055825280, 581256000, 129168000},

{1, 4598, 1184922, 92796080, 2442817180, 21051364536, 73606098792, 129668682240, 123157690560, 60493680000, 12098736000}

The infinite sum triangle is:

Table[Sum[p[k, n]*x^k, {k, 0, Infinity}], {n, 0, 10}];

{1},

{1, 1},

{-1, -18, -1},

{1, 179, 179, 1}

{-1, -2224, -8030, -2224, -1},

{1, 20049, 167150, 167150, 20049, 1},

{-1, -5596844, -145462469, -524422080, -360876091, -48653716, 1},

{1, 5596843, 194116185, 885298171, 885298171, 194116185, 5596843, 1},

{-1, -374989392, -25339790572, -207966886768, -400645626534, -207966886768, -25339790572, -374989392, -1}

MATHEMATICA

Clear[p, x, n, k, a]

p[k, 0] := 1; p[k, 1] := 1 + 2*k;

p[k_, n_] := If[Mod[n, 2] == 0, (1 + 2*k)*p[k, n - 1] + n*Binomial[n + 1, n - 1]*k*(k + 1)*p[k, n - 2], (1 + 2*k)*(1 + 3*(p[k, n - 1] - 1))];

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

Flatten[%]

CROSSREFS

Sequence in context: A193727 A138098 A163914 * A151504 A151507 A151363

Adjacent sequences:  A167880 A167881 A167882 * A167884 A167885 A167886

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Nov 14 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 April 21 12:08 EDT 2014. Contains 240824 sequences.