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%I M4029 N1672 #47 Dec 04 2017 17:46:39
%S 1,5,205,22265,4544185,1491632525,718181418565,476768795646785,
%T 417370516232719345,465849831125196593045,645702241048404020542525,
%U 1088120580608731523115639305,2190881346273790815462670984105
%N Multiples of Euler numbers.
%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
%D Glaisher, J. W. L.; Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002438/b002438.txt">Table of n, a(n) for n = 1..100</a>
%H P. Bala, <a href="/A002438/a002438.pdf">A triangle for calculating A002438</a>
%H P. Bala, <a href="/A002439/a002439.pdf">Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)</a>
%H Matthieu Josuat-Vergès and Jang Soo Kim, <a href="http://arxiv.org/abs/1101.5608">Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity</a>, arXiv:1101.5608 [math.CO] (2011)
%F a(n) = A000364(n-1) * (9^(n-1) + 1)/2.
%F a(n+1) = Sum_{k = 0..n} A086646(n, k)*(-4)^k*9^(n-k). - _Philippe Deléham_, Aug 26 2005
%F From _Peter Bala_, Mar 13 2015: (Start)
%F a(n+1) = (-1)^n*6^(2*n)*E(2*n,1/6).
%F Assuming an offset of 0, the e.g.f. is cos(2*x)/cos(3*x) = 1 + 5*x + 205*x^2/2! + 22265*x^3/3! + 4544185*x^4/4! + ....
%F O.g.f. as a continued fraction: x/(1 - (3^2 - 2^2)*x/(1 - 6^2*x/(1 - (9^2 - 2^2)*x/(1 - 12^2*x/(1 - ... ))))) = x + 5*x^2 + 205*x^3 + 22265*x^4 + 4544185*x^5 + .... See Josuat-Vergès and Kim, p. 23. Cf. A086646.
%F The expansion of exp( Sum_{n >= 1} a(n+1)*x^n/n ) = exp( 5*x + 205*x^2/2 + 22265*x^3/3 + 4544185 *x^4/4 + ... ) appears to have integer coefficients. See A255884.
%F (End)
%F From _Peter Bala_, Nov 10 2015: (Start)
%F O.g.f. A(x) = 1/(1 + x - 6*x/(1 - 30*x/(1 + x - 84*x/(1 - 132*x/(1 + x - ... - 6*n*(6*n - 5)*x/(1 - 6*n*(6*n - 1)*x/(1 + x - ))))))).
%F A(x) = 1/(1 + 25*x - 30*x/(1 - 6*x/(1 + 25*x - 132*x/(1 - 84*x/(1 + 25*x - ... - 6*n*(6*n - 1)*x/(1 - 6*n*(6*n - 5)*x/(1 + 25*x - ))))))). (End)
%t a[n_] := (1+9^(n-1))*Abs[EulerE[2*(n-1)]]/2; Table[ a[n], {n, 1, 13}](* _Jean-François Alcover_, Feb 10 2012 *)
%o (PARI) A002438(n)=A000364(n-1)*(9^(n-1)+1)\2 \\ - _M. F. Hasler_, Jul 21 2013
%Y Cf. A000364, A086646, A255884.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Herman P. Robinson_
%E More terms from _Jon E. Schoenfield_, May 09 2010
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