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 A156222 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 1

%I

%S -2,-6,9,-18,21,-15,-54,57,-51,375,-162,165,-159,1131,4666413,-486,

%T 489,-483,3399,98015025,148865383434975,-1458,1461,-1455,10203,

%U 2058376701,46892624598373299,83234757492356072395126701,-4374,4377,-4371,30615

%N 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

%C This result is an attempt to get the Carlitz q-Euler number recursion to work at q=2.

%C Row sums are:

%C {-2, 3, -12, 327, 4667388, 148865481452919, 83234757539248699051885452,

%C 6403107722784357842299544181680812061276247,

%C 533167131870041204624565122306522559603976943838556766587936111548,

%C 379814469970935772396967354473544697867037238712728549478618581331342218457211 097728768031486199,

%C 18372455397191678019687937935475502510673821112760290710753555455473664707268317382457368748481529925946011336353217348256796830858732,...}

%D 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

%F p(x,n)=Product[(1 - x^k)/(1 - x), {k, 1, n}];

%F Q(x, n) = (-1/q)*Q[x - 1, n] + (p[q, 2]/q)*p[2, x - 1]^n

%e {-2},

%e {-6, 9},

%e {-18, 21, -15},

%e {-54, 57, -51, 375},

%e {-162, 165, -159, 1131, 4666413},

%e {-486, 489, -483, 3399, 98015025, 148865383434975},

%e {-1458, 1461, -1455, 10203, 2058376701, 46892624598373299, 83234757492356072395126701},

%e {-4374, 4377, -4371, 30615, 43226094369, 14771177353650155631, 812787407355992348787877344369, 6403107722783545054892173418154627307655631},

%e {-13122, 13125, -13119, 91851, 907748532813, 4652920879108270217187, 7936869032970852911892907767282813, 3939159855518315085924426024841736611828307717187, 533167131870041200685405266788199536810517943490987381835842282813},

%t Clear[Q, e, p, n, x];

%t p[x_, n_] := Product[(1 - x^k)/(1 - x), {k, 1, n}];

%t q = 2; e[0] = 1; e[n_] := e[n] = -q*(q*e[0] + 1)^n; Table[e[n], {n, 0, 30}];

%t Q[0, n] := e[n]; Q[x, 0] := 1;

%t Q[x_, n_] := Q[x, n] = (-1/q)*Q[x - 1, n] + (p[q, 2]/q)*p[2, x - 1]^n;

%t a0 = Table[Table[ExpandAll[2^x*Q[x, n]], {x, 0, m}] /. n -> m, {m, 0, 10}];

%t Flatten[%]

%K uned,sign

%O 0,1

%A _Roger L. Bagula_, Feb 06 2009

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Last modified October 16 15:51 EDT 2019. Contains 328101 sequences. (Running on oeis4.)