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A158782 Skip power infinite sum polynomials: p(x,n)=((1 - x)^ (n + 1))*((1 + x)^ (n + 1))*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity, 2}]. 0
1, 1, 0, 3, 1, 0, 22, 0, 9, 1, 0, 121, 0, 235, 0, 27, 1, 0, 620, 0, 3446, 0, 1996, 0, 81, 1, 0, 3119, 0, 40314, 0, 63854, 0, 15349, 0, 243, 1, 0, 15618, 0, 422087, 0, 1434812, 0, 963327, 0, 112546, 0, 729, 1, 0, 78117, 0, 4157997, 0, 26672209, 0, 37898739, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are:

{1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720,

3805072588800, ...}.

This polynomial set is the skip power version of the infinite sum for MacMahon A060187.

If we call the odd version: q(x,n)=((1 - x)^ (n + 1))*((1 + x)^ (n + 1))*Sum[(2*k + 1)^n*x^k, {k, 1, Infinity, 2}].

Then the sum:

r(x,n)=p(x,n)+q(x,n)

is symmetrical and double wide:

r(x,n)=(x+1)^(2*n+1)*A060187(x,n)

LINKS

Table of n, a(n) for n=0..58.

FORMULA

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

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

EXAMPLE

{1},

{1, 0, 3},

{1, 0, 22, 0, 9},

{1, 0, 121, 0, 235, 0, 27},

{1, 0, 620, 0, 3446, 0, 1996, 0, 81},

{1, 0, 3119, 0, 40314, 0, 63854, 0, 15349, 0, 243},

{1, 0, 15618, 0, 422087, 0, 1434812, 0, 963327, 0, 112546, 0, 729},

{1, 0, 78117, 0, 4157997, 0, 26672209, 0, 37898739, 0, 12960063, 0, 806047, 0, 2187},

{1, 0, 390616, 0, 39531132, 0, 442372648, 0, 1151050534, 0, 840642408, 0, 162711868, 0, 5705752, 0, 6561},

{1, 0, 1953115, 0, 367889284, 0, 6818184988, 0, 29742429982, 0, 39523450714, 0, 16677432820, 0, 1955297356, 0, 40156777, 0, 19683},

{1, 0, 9765614, 0, 3379362581, 0, 100040972648, 0, 689712304370, 0, 1514068354580, 0, 1167881384066, 0, 306865115624, 0, 22833444557, 0, 281825710, 0, 59049}

MATHEMATICA

Clear[p, x, n, m];

p[x_, n_] = ((1 - x)^ (n + 1))*((1 + x)^ (n + 1))*Sum[(2*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

Sequence in context: A246049 A316773 A006837 * A187558 A327547 A233293

Adjacent sequences:  A158779 A158780 A158781 * A158783 A158784 A158785

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 26 2009

STATUS

approved

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)