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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

L. Carlitz, Some arithmetic properties of the Olivier functions, Math. Ann. 128, (1954) 412-419.

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.

LINKS

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

H. J. Haubold, A. M. Mathai, R. K. Saxena, Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, vol. 2011.

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: A089944 A180165 A142249 * A257243 A097351 A207330

Adjacent sequences:  A274702 A274703 A274704 * A274706 A274707 A274708

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jul 03 2016

STATUS

approved

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Last modified November 21 03:01 EST 2018. Contains 317427 sequences. (Running on oeis4.)