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 A002438 Multiples of Euler numbers. (Formerly M4029 N1672) 6
 1, 5, 205, 22265, 4544185, 1491632525, 718181418565, 476768795646785, 417370516232719345, 465849831125196593045, 645702241048404020542525, 1088120580608731523115639305, 2190881346273790815462670984105 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES 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. Glaisher, J. W. L.; Messenger of Math., 28 (1898), 36-79, see esp. p. 51. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..100 P. Bala, A triangle for calculating A002438 Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO] (2011) FORMULA a(n) = A000364(n-1) * (9^(n-1) + 1)/2. a(n+1) = Sum_{k = 0..n} A086646(n, k)*(-4)^k*9^(n-k). - Philippe Deléham, Aug 26 2005 From Peter Bala, Mar 13 2015: (Start) a(n+1) = (-1)^n*6^(2*n)*E(2*n,1/6). 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! + .... 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. 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. (End) From Peter Bala, Nov 10 2015: (Start) 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 - ))))))). 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) MATHEMATICA a[n_] := (1+9^(n-1))*Abs[EulerE[2*(n-1)]]/2; Table[ a[n], {n, 1, 13}](* Jean-François Alcover, Feb 10 2012 *) PROG (PARI) A002438(n)=A000364(n-1)*(9^(n-1)+1)\2 \\ - M. F. Hasler, Jul 21 2013 CROSSREFS Cf. A000364, A086646, A255884. Sequence in context: A012811 A186722 A060486 * A005333 A162087 A292330 Adjacent sequences:  A002435 A002436 A002437 * A002439 A002440 A002441 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Herman P. Robinson More terms from Jon E. Schoenfield, May 09 2010 STATUS approved

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)