A274705 Rectangular array read by ascending antidiagonals. Row n has the exponential generating function 1/M_{n}(z^n) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only, for n>=1.

1, 1, -2, 1, -3, 3, 1, -4, 25, -4, 1, -5, 133, -427, 5, 1, -6, 621, -15130, 12465, -6, 1, -7, 2761, -437593, 4101799, -555731, 7, 1, -8, 11999, -12012016, 1026405753, -2177360656, 35135945, -8, 1, -9, 51465, -325204171, 243458990271, -6054175060941, 1999963458217, -2990414715, 9

Offset: 0

Author

Peter Luschny, Jul 03 2016

Keywords

• sign
• tabl

Examples

			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, ...}
		

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.

Crossrefs

Cf. A009843, A181983, A274703, A274704.

Programs

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}]

Sage

def ibn(m, k):
    w = exp(2*pi*I/m)
    om = m*x/sum(exp(x*w^j) for j in range(m))
    t = taylor(om, x, 0, k + m)
    return simplify(factorial(k)*t.list()[k])
def A274705_row(m, size):
    return [ibn(m, k) for k in range(1, m*size, m)]
for n in (1..4): print(A274705_row(n, 8))

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).