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A154337 A triangular sequence of coefficients of polynomials: p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2. 2
1, 1, 1, 1, 7, 1, 1, 29, 29, 1, 1, 101, 312, 101, 1, 1, 327, 2372, 2372, 327, 1, 1, 1023, 15219, 34114, 15219, 1023, 1, 1, 3145, 88839, 381775, 381775, 88839, 3145, 1, 1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1, 1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 9, 60, 516, 5400, 66600, 947520, 15301440, 276877440,...}

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1274

FORMULA

p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.

Functional form:

p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x)/2.

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

EXAMPLE

{1},

{1, 1},

{1, 7, 1},

{1, 29, 29, 1},

{1, 101, 312, 101, 1},

{1, 327, 2372, 2372, 327, 1},

{1, 1023, 15219, 34114, 15219, 1023, 1},

{1, 3145, 88839, 381775, 381775, 88839, 3145, 1},

{1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1},

{1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1}

MATHEMATICA

Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]

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

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A168517 A142465 A248829 * A033933 A108267 A156916

Adjacent sequences:  A154334 A154335 A154336 * A154338 A154339 A154340

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Jan 07 2009

STATUS

approved

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Last modified November 18 15:41 EST 2018. Contains 317323 sequences. (Running on oeis4.)