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

0,3

Links

Table of n, a(n) for n=0..44.

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

(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))

Crossrefs

Cf. A009843, A181983, A274703, A274704.

Sequence in context: A180165 A358349 A142249 * A257243 A097351 A207330

Adjacent sequences: A274702 A274703 A274704 * A274706 A274707 A274708

Keyword

sign,tabl

Author

Peter Luschny, Jul 03 2016

Status

approved