OFFSET
0,3
LINKS
L. Carlitz, Some arithmetic properties of the Olivier functions, Math. Ann. 128 (1954), 412-419.
H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
Eric Weisstein's MathWorld, Generalized hyperbolic functions.
FORMULA
Recurrence for the m-th row: R(m, n) = -Sum_{k=0..n-1} binomial(m*n+1, m*k+1)*R(m, k) for n >= 1. See Carlitz (1.3).
EXAMPLE
Array starts:
n=1: {1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11,...} [A181983]
n=2: {1, -3, 25, -427, 12465, -555731, 35135945,...} [A009843]
n=3: {1, -4, 133, -15130, 4101799, -2177360656,...} [A274703]
n=4: {1, -5, 621, -437593, 1026405753, -6054175060941,...} [A274704]
n=5: {1, -6, 2761, -12012016, 243458990271, ...}
MAPLE
ibn := proc(m, k) local w, om, t;
w := exp(2*Pi*I/m); om := m*x/add(exp(x*w^j), j=0..m-1);
t := series(om, x, k+m); simplify(k!*coeff(t, x, k)) end:
seq(seq(ibn(n-k+2, n*k-n-k^2+3*k-1), k=1..n+1), n=0..8);
MATHEMATICA
A274705Row[m_] := Module[{c}, c = CoefficientList[Series[1/MittagLefflerE[m, z^m],
{z, 0, 12*m}], z]; Table[Factorial[m*n+1]*c[[m*n+1]], {n, 0, 9}] ]
Table[Print[A274705Row[n]], {n, 1, 6}]
PROG
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jul 03 2016
STATUS
approved