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A142158 An infinite sum polynomial triangular sequence of coefficients that gives a LerchPhi polynomial: p(x,n)=(1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]=(1+x)^n*LerchPhi[x,-n,n]. 0
0, 1, 1, 4, -3, 1, 27, -44, 31, -8, 256, -655, 731, -389, 81, 3125, -10974, 17026, -13934, 5901, -1024, 46656, -208943, 418377, -465898, 300182, -105279, 15625, 823543, -4491192, 11064957, -15661904, 13617801, -7229592, 2161363, -279936, 16777216, -107948223, 316559287, -545245307, 598756419 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The row sums are n!.

LINKS

Table of n, a(n) for n=1..41.

FORMULA

p(x,n)=(1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]=(1+x)^n*LerchPhi[x,-n,n]; t(n,m)=coefficients(p(x,n)).

EXAMPLE

{0, 1},

{1},

{4, -3, 1},

{27, -44,31, -8},

{256, -655, 731, -389, 81},

{3125, -10974, 17026, -13934, 5901, -1024},

{46656, -208943, 418377, -465898, 300182, -105279, 15625},

{823543, -4491192, 11064957, -15661904,13617801, -7229592, 2161363, -279936},

{16777216, -107948223, 316559287, -545245307, 598756419, -427227197, 192806917, -50203593, 5764801},

{387420489, -2874204890, 9791869696, -19910155238, 26472644638, -23777517254, 14389038880, -5646339386, 1301823673, -134217728},

{10000000000, -84062575399, 326605693613, -766674161560, 1198591217792, -1299948741046, 988352227754, -519310387408, 180244457240, -37280886587, 3486784401}

MATHEMATICA

Clear[p, x, n]; p[x_, n_] = (1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]; Table[FullSimplify[Expand[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

CROSSREFS

Sequence in context: A298673 A245732 A039621 * A203412 A217756 A154960

Adjacent sequences:  A142155 A142156 A142157 * A142159 A142160 A142161

KEYWORD

sign,uned

AUTHOR

Roger L. Bagula, Sep 16 2008

STATUS

approved

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Last modified May 7 12:52 EDT 2021. Contains 343650 sequences. (Running on oeis4.)