A251592 Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.

1, 0, 2, 0, -3, 9, 0, 8, -48, 64, 0, -30, 275, -750, 625, 0, 144, -1800, 7560, -12960, 7776, 0, -840, 13426, -77175, 204085, -252105, 117649, 0, 5760, -112896, 831488, -3010560, 5734400, -5505024, 2097152, 0, -45360, 1058508, -9573228

Offset

1,3

Comments

Second column (unsigned) 2, 3, 8, 30, 144, ... is A001048.

Diagonal 1, 2, 9, 64, 625, 7776, ... is A000169.

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

Links

Table of n, a(n) for n=1..40.

Peter Bala, Fractional iteration of a series inversion operator

MathOverflow, Properties and name of some polynomials

Eric Weisstein's MathWorld, Mittag-Leffler Function

Eric Weisstein's MathWorld, Wright Function

Formula

P(n,t) = (n-1)!*binomial(n*t, n-1).

From Peter Bala, Nov 15 2015: (Start)

E.g.f. (with constant term 1): B_t(x) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + 1,n)*x^n = 1 + x + 2*t*x^2/2! + 3*t(3*t - 1)*x^3/3! + 4*t*(4*t - 1)*(4*t - 2)*x^4/4! + ... is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.

In the notation of the Bala link, B_t(x) = I^t(1 + x) where I^t is a fractional inversion operator. B_(1+t)(x) is the e.g.f. for A260687.

B_t(x) = 1 + x*B_t(x)^t.

For complex r, B_t(x)^r = Sum_{n >= 0} r/(n*t + r)*binomial(n*t + r,n)*x^n.

log (B_t(x)) = Sum_{n >= 1} 1/(n*t)*binomial(n*t,n)*x^n.

B_2(x) is the o.g.f. for the Catalan numbers A000108. B_t(x) for t = 3,4,5,... gives the o.g.f. for various Fuss-Catalan sequences. See the cross references. (End)

Example

Triangle begins :
  1;
  0,   2;
  0,  -3,     9;
  0,   8,   -48,   64;
  0, -30,   275, -750,    625;
  0, 144, -1800, 7560, -12960, 7776;
  ...

Mathematica

P[n_, t_] := Product[n*t - k, {k, 0, n-2}]; row[n_] := CoefficientList[P[n, t], t]; Table[row[n], {n, 1, 10}] // Flatten

Crossrefs

Cf. A000169, A001048, A156136, A000108 (B_2(x)), A001764 (B_3(x)), A002293 (B_4(x)), A002294 (B_5(x)), A002295 (B_6(x)), A002296 (B_7(x)), A007556 (B_8(x)), A062994 (B_9(x)), A059968 (B_10(x)), A230388 (B_11(x)), A139526, A260687.

Sequence in context: A336326 A255384 A236396 * A365985 A327244 A325816

Adjacent sequences: A251589 A251590 A251591 * A251593 A251594 A251595

Keyword

sign,tabl

Author

Jean-François Alcover, Dec 05 2014

Status

approved