|
|
A000795
|
|
Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.
(Formerly M2044 N0810)
|
|
19
|
|
|
1, 2, 12, 152, 3472, 126752, 6781632, 500231552, 48656756992, 6034272215552, 929327412759552, 174008703107274752, 38928735228629389312, 10255194381004799025152, 3142142941901073853366272, 1107912434323301224813002752, 445427836895850552387642130432
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Named after the German mathematician Hans Salié (1902-1978). -Amiram Eldar,Jun 10 2021
|
|
REFERENCES
|
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 86, Problem 32.
Hans Salié, Arithmetische Eigenschaften der Koeffizienten einer speziellen Hurwitzschen Potenzreihe, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Natur. Reihe 12 (1963), pp. 617-618.
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
|
|
|
FORMULA
|
G.f.: 1 / (1 - (1^2+1)*x / (1 - 2^2*x / (1 - (3^2+1)*x / (1 - 4^2*x / (1 - (5^2+1)*x / (1 - 6^2*x /...)))))). -Michael Somos,May 12 2012
G.f.: Q(0)/(1-2*x), where Q(k) = 1 - 8*x^2*(2*k^2+2*k+1)*(k+1)^2/( 8*x^2*(2*k^2+2*k+1)*(k+1)^2 - (1 - 8*x*k^2 - 4*x*k -2*x)*(1 - 8*x*k^2 - 20*x*k -14*x)/Q(k+1) ); (continued fraction). -Sergei N. Gladkovskii,Oct 22 2013
a(n) ~ (2*n)! * 2^(2*n+2) * cosh(Pi/2) / Pi^(2*n+1). -Vaclav Kotesovec,Mar 08 2014
a(n) = 1 - Sum_{k=1..n} (-1)^k * binomial(2*n,2*k) * a(n-k). -Ilya Gutkovskiy,Mar 09 2022
|
|
EXAMPLE
|
cosh x / cos x = Sum_{n>=0} a(n)*x^(2n)/(2n)! = 1 + x^2 + (1/2)*x^4 + (19/90)*x^6 + (31/360)*x^8 + (3961/113400)*x^10 +...
G.f. = 1 + 2*x + 12*x^2 + 252*x^3 + 3472*x^4 + 126752*x^5 + 6781632*x^6 +...
|
|
MAPLE
|
(2*n)!*coeftayl( cosh(x)/cos(x), x=0, 2*n);
|
|
MATHEMATICA
|
max = 16; se = Series[ Cosh[x] / Cos[x], {x, 0, 2*max} ]; a[n_]:= SeriesCoefficient[ se, 2*n ]*(2*n)!; Table[ a[n], {n, 0, max} ] (*Jean-François Alcover,Apr 02 2012 *)
With[{nn=40}, Take[CoefficientList[Series[Cosh[x]/Cos[x], {x, 0, nn}], x] Range[ 0, nn]!, {1, -1, 2}]] (*Harvey P. Dale,May 11 2012 *)
a[ n_]:= If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ Cosh[ x] / Cos[ x], {x, 0, m}]]]; (*Michael Somos,Aug 15 2015 *)
|
|
PROG
|
(Sage) # Generalized algorithm of L. Seidel (1877)
R = []; A = {-1:0, 0:0}
k = 0; e = 1
for i in range(n):
Am = 1 if e == 1 else 0
A[k + e] = 0
e = -e
for j in (0..i):
Am += A[k]
A[k] = Am
k += e
if e == -1: R.append(A[-i//2])
return R
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|