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A178232 A triangle sequence derived from setting a Euler numbers A122045 generalization equal to the Eulerian numbers A008292 to get a generating function expansion: p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(t*x) + t - exp(t)* x)). 1

%I

%S 0,0,1,6,1,1,36,8,3,7,1,240,60,-20,81,11,21,1,1800,480,-510,822,143,

%T 173,123,51,1,15120,4200,-7560,8526,2450,239,2381,435,715,113,1,

%U 141120,40320,-102480,93744,43512,-21320,36991,2943,11035,4035,3139,239,1

%N A triangle sequence derived from setting a Euler numbers A122045 generalization equal to the Eulerian numbers A008292 to get a generating function expansion: p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(t*x) + t - exp(t)* x)).

%C The first column gives the Lah numbers A001286:(n - 1)*n!/2;

%C {0,0,1, 6, 36, 240, 1800, 15120, 141120, 1451520, ...}

%C Row sums are {0, 0, 1, 8, 55, 394, 3083, 26620, 253279, 2642390, 30052699, ...}.

%C The equation solved in the integer q was

%C q*exp(x*t)/(q - 1 + exp(t)) - (1 - t)/(1 - t*exp(x*(1 - t))) = 0.

%C Factors and the n! first term from taken out in Mathematica to give a more simple set of coefficients.

%C The idea in solving for an integer q here is to get a polynomial that behaves as a generalization of both types.

%C No q-form value for q=n=0,1 is expected.

%D Steve Roman , The Umbral Calculus, Dover Publications, New York (1984), pp. 78 - 79

%D L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245

%H L. Carlitz, <a href="https://projecteuclid.org/euclid.dmj/1077475200">q-Bernoulli numbers and polynomials</a>, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.

%F p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(t*x) + t - exp(t)* x)).

%e {0},

%e {0},

%e {1},

%e {6, 1, 1},

%e {36, 8, 3, 7, 1},

%e {240, 60, -20, 81, 11, 21, 1},

%e {1800, 480, -510, 822, 143, 173, 123, 51, 1},

%e {15120, 4200, -7560, 8526, 2450, 239, 2381, 435, 715, 113, 1},

%e {141120, 40320, -102480, 93744, 43512, -21320, 36991, 2943, 11035, 4035, 3139, 239, 1},

%e {1451520, 423360, -1391040, 1103760, 763056, -585432, 527544, 71353, 82513, 107377, 39589, 36349, 11947, 493, 1},

%e {16329600, 4838400, -19504800, 13940640, 13361040, -12088080, 7137270, 2643650, -749001, 2527719, 165459, 900099, 256743, 251073, 41883, 1003, 1}

%t p[t_] = ((-1 + Exp[x]) (-1 + x)/(-1 + Exp[t*x] + t - Exp[t]* x));

%t a = Table[ CoefficientList[FullSimplify[ExpandAll[(FullSimplify[ExpandAll[ -(1/((-1 + Exp[x])*(-1 + x)))*x^(n + 1)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]] - n!)/(x^2*(-1 + x))]], x], {n, 0, 10}] Flatten[a]

%Y Cf. A008292, A122045, A156222.

%K sign,tabf,uned

%O 0,4

%A _Roger L. Bagula_, May 23 2010

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Last modified September 20 08:13 EDT 2019. Contains 327214 sequences. (Running on oeis4.)