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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.

%I A274705 #42 Feb 16 2025 08:33:36
%S A274705 1,1,-2,1,-3,3,1,-4,25,-4,1,-5,133,-427,5,1,-6,621,-15130,12465,-6,1,
%T A274705 -7,2761,-437593,4101799,-555731,7,1,-8,11999,-12012016,1026405753,
%U A274705 -2177360656,35135945,-8,1,-9,51465,-325204171,243458990271,-6054175060941,1999963458217,-2990414715,9
%N 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.
%H A274705 L. Carlitz, <a href="https://doi.org/10.1007/BF01360145">Some arithmetic properties of the Olivier functions</a>, Math. Ann. 128 (1954), 412-419.
%H A274705 H. J. Haubold, A. M. Mathai, and R. K. Saxena, <a href="http://dx.doi.org/10.1155/2011/298628">Mittag-Leffler Functions and Their Applications</a>, Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages.
%H A274705 L. Olivier, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0002?tify=%7B%22pages%22:[257]%7D">Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus</a>, J. Reine Angew. Math. 2 (1827), 243-251.
%H A274705 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html">Generalized hyperbolic functions</a>.
%F A274705 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).
%e A274705 Array starts:
%e A274705 n=1: {1, -2,  3, -4, 5, -6, 7, -8,  9, -10,  11,...} [A181983]
%e A274705 n=2: {1, -3, 25, -427, 12465, -555731, 35135945,...} [A009843]
%e A274705 n=3: {1, -4, 133, -15130, 4101799,  -2177360656,...} [A274703]
%e A274705 n=4: {1, -5, 621, -437593, 1026405753, -6054175060941,...} [A274704]
%e A274705 n=5: {1, -6, 2761, -12012016, 243458990271, ...}
%p A274705 ibn := proc(m, k) local w, om, t;
%p A274705 w := exp(2*Pi*I/m); om := m*x/add(exp(x*w^j), j=0..m-1);
%p A274705 t := series(om, x, k+m); simplify(k!*coeff(t,x,k)) end:
%p A274705 seq(seq(ibn(n-k+2, n*k-n-k^2+3*k-1), k=1..n+1),n=0..8);
%t A274705 A274705Row[m_] := Module[{c}, c = CoefficientList[Series[1/MittagLefflerE[m,z^m],
%t A274705 {z,0,12*m}],z]; Table[Factorial[m*n+1]*c[[m*n+1]], {n,0,9}] ]
%t A274705 Table[Print[A274705Row[n]], {n,1,6}]
%o A274705 (Sage)
%o A274705 def ibn(m, k):
%o A274705     w = exp(2*pi*I/m)
%o A274705     om = m*x/sum(exp(x*w^j) for j in range(m))
%o A274705     t = taylor(om, x, 0, k + m)
%o A274705     return simplify(factorial(k)*t.list()[k])
%o A274705 def A274705_row(m, size):
%o A274705     return [ibn(m, k) for k in range(1, m*size, m)]
%o A274705 for n in (1..4): print(A274705_row(n, 8))
%Y A274705 Cf. A009843, A181983, A274703, A274704.
%K A274705 sign,tabl
%O A274705 0,3
%A A274705 _Peter Luschny_, Jul 03 2016