A178232 A triangle sequence derived from setting an 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)).

0, 0, 1, 6, 1, 1, 36, 8, 3, 7, 1, 240, 60, -20, 81, 11, 21, 1, 1800, 480, -510, 822, 143, 173, 123, 51, 1, 15120, 4200, -7560, 8526, 2450, 239, 2381, 435, 715, 113, 1, 141120, 40320, -102480, 93744, 43512, -21320, 36991, 2943, 11035, 4035, 3139, 239, 1

Offset: 0

Author

Roger L. Bagula, May 23 2010

Keywords

• sign
• tabf
• uned

Comments

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

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

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

The equation solved in the integer q was

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

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

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

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

Examples

			{0},
{0},
{1},
{6, 1, 1},
{36, 8, 3, 7, 1},
{240, 60, -20, 81, 11, 21, 1},
{1800, 480, -510, 822, 143, 173, 123, 51, 1},
{15120, 4200, -7560, 8526, 2450, 239, 2381, 435, 715, 113, 1},
{141120, 40320, -102480, 93744, 43512, -21320, 36991, 2943, 11035, 4035, 3139, 239, 1},
{1451520, 423360, -1391040, 1103760, 763056, -585432, 527544, 71353, 82513, 107377, 39589, 36349, 11947, 493, 1},
{16329600, 4838400, -19504800, 13940640, 13361040, -12088080, 7137270, 2643650, -749001, 2527719, 165459, 900099, 256743, 251073, 41883, 1003, 1}
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 78-79.
• L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245.

Links

• L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.

Crossrefs

Cf. A008292, A122045, A156222.

Programs

Mathematica

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

Formula

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