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Search: id:A156222
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| A156222 |
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A triangle sequence of the Carlitz q-Euler number type: q=2;p(x,n)=Product[(1 - x^k)/(1 - x), {k, 1, n}]; Q(x, n) = (-1/q)*Q[x - 1, n] + (p[q, 2]/q)*p[2, x - 1]^n |
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+0
1
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| -2, -6, 9, -18, 21, -15, -54, 57, -51, 375, -162, 165, -159, 1131, 4666413, -486, 489, -483, 3399, 98015025, 148865383434975, -1458, 1461, -1455, 10203, 2058376701, 46892624598373299, 83234757492356072395126701, -4374, 4377, -4371, 30615
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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This result is an attempt to get the Carlitz q-Euler number recursion to work at q=2.
Row sums are:
{-2, 3, -12, 327, 4667388, 148865481452919, 83234757539248699051885452,
6403107722784357842299544181680812061276247,
533167131870041204624565122306522559603976943838556766587936111548,
379814469970935772396967354473544697867037238712728549478618581331342218457211 097728768031486199,
18372455397191678019687937935475502510673821112760290710753555455473664707268317382457368748481529925946011336353217348256796830858732,...}
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REFERENCES
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L. Carlitz,q-Bernoulli numbers and polynomials,Duke Math. J. Volume 15, Number 4 (1948), 987-1000.http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077475200
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FORMULA
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p(x,n)=Product[(1 - x^k)/(1 - x), {k, 1, n}];
Q(x, n) = (-1/q)*Q[x - 1, n] + (p[q, 2]/q)*p[2, x - 1]^n
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EXAMPLE
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{-2},
{-6, 9},
{-18, 21, -15},
{-54, 57, -51, 375},
{-162, 165, -159, 1131, 4666413},
{-486, 489, -483, 3399, 98015025, 148865383434975},
{-1458, 1461, -1455, 10203, 2058376701, 46892624598373299, 83234757492356072395126701},
{-4374, 4377, -4371, 30615, 43226094369, 14771177353650155631, 812787407355992348787877344369, 6403107722783545054892173418154627307655631},
{-13122, 13125, -13119, 91851, 907748532813, 4652920879108270217187, 7936869032970852911892907767282813, 3939159855518315085924426024841736611828307717187, 533167131870041200685405266788199536810517943490987381835842282813},
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MATHEMATICA
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Clear[Q, e, p, n, x];
p[x_, n_] := Product[(1 - x^k)/(1 - x), {k, 1, n}];
q = 2; e[0] = 1; e[n_] := e[n] = -q*(q*e[0] + 1)^n; Table[e[n], {n, 0, 30}];
Q[0, n] := e[n]; Q[x, 0] := 1;
Q[x_, n_] := Q[x, n] = (-1/q)*Q[x - 1, n] + (p[q, 2]/q)*p[2, x - 1]^n;
a0 = Table[Table[ExpandAll[2^x*Q[x, n]], {x, 0, m}] /. n -> m, {m, 0, 10}];
Flatten[%]
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CROSSREFS
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Sequence in context: A054974 A072481 A032471 this_sequence A002886 A028724 A156190
Adjacent sequences: A156219 A156220 A156221 this_sequence A156223 A156224 A156225
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KEYWORD
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uned,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2009
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