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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 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 January 22 15:57 EST 2019. Contains 319364 sequences. (Running on oeis4.)