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A158781 A doubled power MacMahon A060187 infinite sum polynomial from the Eulerian numbers sum by skipping powers: p(x,n)=((1 - x)^ (n + 1))*((1 + x)^ (n + 1))*Sum[(k + 1)^n*x^k, {k, 0, Infinity, 2}] 0
1, 1, 0, 1, 1, 0, 6, 0, 1, 1, 0, 23, 0, 23, 0, 1, 1, 0, 76, 0, 230, 0, 76, 0, 1, 1, 0, 237, 0, 1682, 0, 1682, 0, 237, 0, 1, 1, 0, 722, 0, 10543, 0, 23548, 0, 10543, 0, 722, 0, 1, 1, 0, 2179, 0, 60657, 0, 259723, 0, 259723, 0, 60657, 0, 2179, 0, 1, 1, 0, 6552, 0, 331612, 0, 2485288 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Row sums are:A000165;2^n*n!;

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

p(Sqrt[x],n) gives the MacMahon triangle sequence: A060187.

This power skipping approach to infinite sums is a new way to look at them.

LINKS

Table of n, a(n) for n=0..70.

FORMULA

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

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

EXAMPLE

{1},

{1, 0, 1},

{1, 0, 6, 0, 1},

{1, 0, 23, 0, 23, 0, 1},

{1, 0, 76, 0, 230, 0, 76, 0, 1},

{1, 0, 237, 0, 1682, 0, 1682, 0, 237, 0, 1},

{1, 0, 722, 0, 10543, 0, 23548, 0, 10543, 0, 722, 0, 1}, {

1, 0, 2179, 0, 60657, 0, 259723, 0, 259723, 0, 60657, 0, 2179, 0, 1},

{1, 0, 6552, 0, 331612, 0, 2485288, 0, 4675014, 0, 2485288, 0, 331612, 0, 6552, 0, 1},

{1, 0, 19673, 0, 1756340, 0, 21707972, 0, 69413294, 0, 69413294, 0, 21707972, 0, 1756340, 0, 19673, 0, 1},

{1, 0, 59038, 0, 9116141, 0, 178300904, 0, 906923282, 0, 1527092468, 0, 906923282, 0, 178300904, 0, 9116141, 0, 59038, 0, 1}

MATHEMATICA

Clear[p, x, n, m];

p[x_, n_] = ((1 - x)^ (n + 1))*((1 + x)^ (n + 1))*Sum[(k + 1)^n*x^k, {k, 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

A060187, A000165

Sequence in context: A134899 A076413 A154305 * A293299 A293486 A195398

Adjacent sequences:  A158778 A158779 A158780 * A158782 A158783 A158784

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2009

STATUS

approved

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Last modified June 22 18:49 EDT 2021. Contains 345388 sequences. (Running on oeis4.)