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Search: id:A142158
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| A142158 |
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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]. |
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+0
1
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| 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; listen)
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OFFSET
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1,4
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COMMENT
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The row sums are n!.
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FORMULA
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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)).
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EXAMPLE
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{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}
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MATHEMATICA
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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[%]
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CROSSREFS
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Sequence in context: A098234 A128320 A039621 this_sequence A154960 A143543 A067017
Adjacent sequences: A142155 A142156 A142157 this_sequence A142159 A142160 A142161
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KEYWORD
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sign,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2008
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