A002438 Multiples of Euler numbers. (Formerly M4029 N1672)

1, 5, 205, 22265, 4544185, 1491632525, 718181418565, 476768795646785, 417370516232719345, 465849831125196593045, 645702241048404020542525, 1088120580608731523115639305, 2190881346273790815462670984105

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

P. Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)

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

N. J. A. Sloane

Extensions

More terms from Herman P. Robinson

More terms from Jon E. Schoenfield, May 09 2010

Status

approved