A251592 - OEIS

%I #24 Feb 15 2024 04:32:32

%S 1,0,2,0,-3,9,0,8,-48,64,0,-30,275,-750,625,0,144,-1800,7560,-12960,

%T 7776,0,-840,13426,-77175,204085,-252105,117649,0,5760,-112896,831488,

%U -3010560,5734400,-5505024,2097152,0,-45360,1058508,-9573228

%N 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.

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

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

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

%H Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>

%H MathOverflow, <a href="http://mathoverflow.net/questions/188617">Properties and name of some polynomials</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Mittag-LefflerFunction.html">Mittag-Leffler Function</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/WrightFunction.html">Wright Function</a>

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

%F From _Peter Bala_, Nov 15 2015: (Start)

%F 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.

%F 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.

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

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

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

%F 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)

%e Triangle begins :

%e   1;

%e   0,   2;

%e   0,  -3,     9;

%e   0,   8,   -48,   64;

%e   0, -30,   275, -750,    625;

%e   0, 144, -1800, 7560, -12960, 7776;

%e   ...

%t 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

%Y 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.

%K sign,tabl

%O 1,3

%A _Jean-François Alcover_, Dec 05 2014