A274704
Exponential generating function 1/M_{4}(z^4) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only.
Original entry on oeis.org
1, -5, 621, -437593, 1026405753, -6054175060941, 75477454065058725, -1766732850877953050849, 71248914440011028226682737, -4637564239713542128355021380117, 462852368857623061805761137170608989, -67965094887205237792816627191801312013545
Offset: 0
-
s := series(2*z/(cosh(z)+cos(z)),z,60):
seq((4*n+1)!*coeff(s,z,4*n+1),n=0..11);
-
c = CoefficientList[Series[1/MittagLefflerE[4, z^4], {z, 0, 15*4}], z];
Table[Factorial[4*n+1]*c[[4*n+1]], {n, 0, 12}]
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.
Original entry on oeis.org
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
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, ...}
- 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.
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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);
-
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}]
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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))
Showing 1-2 of 2 results.