|
| |
|
|
A000364
|
|
Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sech(x)=1/cosh(x).
(Formerly M4019 N1667)
|
|
121
|
|
|
|
1, 1, 5, 61, 1385, 50521, 2702765, 199360981, 19391512145, 2404879675441, 370371188237525, 69348874393137901, 15514534163557086905, 4087072509293123892361, 1252259641403629865468285, 441543893249023104553682821, 177519391579539289436664789665
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Inverse Gudermannian gd^(-1)(x) = log(sec(x) + tan(x)) = log(tan(pi/4 + x/2)) = atanh(sin(x)) = 2 * atanh(tan(x/2)) = 2 * atanh(csc(x) - cot(x)). - Michael Somos, Mar 19 2011
a(n) = number of downup permutations of [2n]. Example: a(2)=5 counts 4231, 4132, 3241, 3142, 2143. - David Callan, Nov 21 2011
a(n) = number of increasing full binary trees on vertices {0,1,2,...,2n} for which the leftmost leaf is labelled 2n. - David Callan, Nov 21 2011
|
|
|
REFERENCES
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810; gives a version with signs: E_{2n} = (-1)^n*a(n) (this is A028296).
J. M. Borwein and D. M. Bailey, Mathematics by Experiment, Peters, Boston, 2004; p. 49
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 141.
J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, Arxiv preprint arXiv:1108.0286, 2011
G. Chrystal, Algebra, Vol. II, p. 342.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
A. L. Edmonds and S, Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396, 2012. - From N. J. A. Sloane, Jan 02 2013
L. Euler, Inst. Calc. Diff., Section 224.
D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
Knuth, D. E.; Buckholtz, Thomas J. Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
R. Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602, 2012. - From N. J. A. Sloane, Jan 25 2013
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
L. Seidel, Ueber eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. - From N. J. A. Sloane, Feb 07 2013
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).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
M. A. Stern, Crelle, 79 (1875), 67-98.
|
|
|
LINKS
|
N. J. A. Sloane, The first 100 Euler numbers: Table of n, a(n) for n = 0..99
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
P. Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - From N. J. A. Sloane, Dec 27 2012
R. Bacher AND P. Flajolet, Pseudo-factorials, elliptic functions, and continued fractions, arXiv:0901.1379.
A. Bucur, J. Lopez-Bonilla, J. Robles-Garcia, A note on the Namias identity for Bernoulli numbers, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, vol.6, no.1, #R21, (1999).
J. Lovejoy and K. Ono, Hypergeometric generating functions for values of Dirichlet and other L-functions, Proc. Nat. Acad. Sci., Vol. 100, No.12, 2003, 6904-6909. [From Peter Bala, Mar 24 2009]
J. Malenfant, Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers, arxiv:1103.1585v6 [math.NT]
Hisanori Mishima, Factorizations of Euler numbers n=0..78, n=80..106.
_Simon Plouffe_, The first 7153 Euler numbers (165 megs)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
C. Radoux, Determinants de Hankel et theoreme de Sylvester
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, A Famous Application of the Encyclopedia of Integer Sequence (Vugraph from a talk about the OEIS)
R. P. Stanley, Alternating permutations and symmetric functions
R. P. Stanley, Permutations
D. C. Vella, Explicit Formulas for Bernoulli and Euler Numbers, Integers 8(1), A1, 2008.
Sam Wagstaff, Prime divisors of the Bernoulli and Euler numbers
Eric Weisstein's World of Mathematics, Euler Number, Secant Number, Alternating Permutation.
Wolfram Research, Generating functions for E_n
Index entries for "core" sequences
Index entries for sequences related to boustrophedon transform
|
|
|
FORMULA
|
E.g.f.: Sum_{n >= 0} a(n) * x^(2*n) / (2*n)! = sec(x). - Michael Somos, Aug 15 2007
E.g.f.: Sum_{n >= 0} a(n) * x^(2*n+1) / (2*n+1)! = gd^(-1)(x). - Michael Somos, Aug 15 2007
E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = 2*arctanh(cosec(x)-cotan(x)). - Ralf Stephan, Dec 16 2004
Pi/4 - [Sum_{k=0..n-1} (-1)^k/(2*k+1)] ~ (1/2)*[Sum_{k>=0} (-1)^k*E(k)/(2*n)^(2k+1)] for positive even n. [Borwein, Borwein, and Dilcher]
Let M_n be the n X n matrix M_n(i, j) = binomial(2*i, 2*(j-1)) = A086645(i, j-1); then for n>0, a(n) = det(M_n); example : det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 . - Philippe DELEHAM,, Sep 04 2005
This sequence is also (-1)^n*EulerE[2*n] or Abs[EulerE[2*n]]. - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
a(n) = 2^n * E_n(1/2), where E_n(x) is an Euler polynomial.
a(k) = a(l) (mod 2^n) if and only if k=l (mod 2^n) (k and l are even). [Stern; see also Wagstaff and Sun]
E_k(3^(k+1)+1)/4=(3^k/2)*sum(j=0..2^n-1, (-1)^(j-1)*(2j+1)^k*[(3j+1)/2^n] (mod 2^n) where k is even and [x] is the greatest integer function. [Sun]
a(n) ~ 2^(n+2)*n!/Pi^(n+1) as n -> infinity.
a(n) = sum(k=0..n, A094665(n, k)*2^(n-k) ). - Philippe Deléham, Jun 10 2004
Recurrence: a(n) = -(-1)^n*sum(i=0..n-1, (-1)^i*a(i)*C(2*n, 2*i) ). - Ralf Stephan, Feb 24 2005
O.g.f.: 1/(1-x/(1-4*x/(1-9*x/(1-16*x/(...-n^2*x/(1-...)))))) (continued fraction). - Paul D. Hanna, Oct 07 2005
a(n)=Integrate[Log[Tan[t/2]^2]^(2n),{t,0,Pi}]/Pi^(2n+1). - Logan Kleinwaks (kleinwaks(AT)alumni.princeton.edu), Mar 15 2007
Contribution from Peter Bala, Mar 24 2009: (Start)
Basic hypergeometric generating function: 2*exp(-t)*Sum {n = 0..inf} Product {k = 1..n} (1-exp(-(4*k-2)*t))*exp(-2*n*t)/Product {k = 1..n+1} (1+exp(-(4*k-2)*t)) = 1 + t + 5*t^2/2! + 61*t^3/3! + .... For other sequences with generating functions of a similar type see A000464, A002105, A002439, A079144 and A158690.
a(n) = 2*(-1)^n*L(-2*n), where L(s) is the Dirichlet L-function L(s) = 1 - 1/3^s + 1/5^s - + .... (End)
sum(n>=0, a(n)*z^(2*n)/(4*n)!! ) = Beta(1/2-z/(2*Pi),1/2+z/(2*Pi))/Beta(1/2,1/2) with Beta(z,w) the Beta function. [Johannes W. Meijer, Jul 06 2009]
a(n)=sum(sum(binomial(k,m)*(-1)^(n+k)/(2^(m-1))*sum(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n), n>0 [From Kruchinin Vladimir, Aug 05 2010]
If n is prime, then a(n)==1 (mod 2*n). [From Vladimir Shevelev, Sep 04 2010]
From Peter Bala: (Start)
(1)... a(n) = (-1/4)^n*B(2*n,-1),
where {B(n,x)}n>=1 = [1,1+x,1+6*x+x^2,1+23*x+23*x^2+x^3,...] is the sequence of Eulerian polynomials of type B - see A060187. Equivalently,
(2)... a(n) = sum {k = 0..2*n} sum {j = 0..k} (-1)^(n-j) *binomial(2*n+1,k-j)*(j+1/2)^(2*n).
We also have
(3)... a(n) = 2*A(2*n,I)/(1+I)^(2*n+1),
where I = sqrt(-1) and where {A(n,x)}n>=1 = [x,x+x^2,x+4*x^2+x^3,...] denotes the sequence of Eulerian polynomials - see A008292. Equivalently,
(4)... a(n) = I*sum {k = 1..2*n} (-1)^(n+k)*k!*Stirling2(2*n,k) *((1+I)/2)^(k-1)
= I*sum {k = 1..2*n} (-1)^(n+k)*((1+I)/2)^(k-1) sum {j = 0..k}
(-1)^(k-j)*binomial(k,j)*j^(2*n).
Either this explicit formula for a(n) or (2) above may be used to obtain congruence results for a(n). For example, for prime p
(5a)... a(p) = 1 (mod p)
(5b)... a(2*p) = 5 (mod p)
and for odd prime p
(6a)... a((p+1)/2) = (-1)^((p-1)/2) (mod p)
(6b)... a((p-1)/2) = -1 + (-1)^((p-1)/2) (mod p).
(End)
It appears that a(n) = (2/Pi)^(2*n+1) * Integrate[EllipticF[x, 1]^(2*n), {x, 0, Pi/2}] (* From Vladimir Reshetnikov, Jan 24 2011 *)
a(n) = (-1)^n*2^(4*n+1)*(zeta(-2*n,1/4)-zeta(-2*n,3/4)) - [Gerry Martens, May 27 2011]
a(n) may be expressed as a sum of multinomials taken over all compositions of 2*n into even parts (Vella 2008): a(n) = sum {compositions 2*i_1+...+2*i_k = 2*n} (-1)^(n+k)* multinom(2*n,2*i_1,...,2*i_k). For example, there are 4 compositions of the number 6 into even parts, namely 6, 4+2, 2+4 and 2+2+2, and hence a(3) = 6!/6!-6!/(4!*2!)-6!/(2!*4!)+6!/(2!*2!*2!) = 61. A companion formula expressing a(n) as a sum of multinomials taken over the compositions of 2*n-1 into odd parts has been given by (Malenfant 2011). - Peter Bala, Jul 07 2011
a(n) = the upper left term in M^n, where M is an infinite square production matrix; M[i,j] = A000290(i) = i^2, i>=1 and 1<=j<=i+1, and M[i,j] = 0, i>=1 and j>=i+2, see the examples. [Gary W. Adamson, Jul 18 2011]
E.g.f.: (sec(x)) = 1+x^2/T(0), T(k) = 2(k+1)(2k+1) - x^2 + x^2*(2k+1)(2k+2)/T(k+1) (continued fraction). - Sergei N. Gladkovskii, Oct 31 2011
E.g.f. A'(x) satisfies the differential equation A'(x)=cos(A(x)). [From Vladimir Kruchinin, Nov 03 2011]
Contribution from Peter Bala, Nov 28 2011: (Begin)
a(n) = D^(2*n)(cosh(x)) evaluated at x = 0, where D is the operator cosh(x)*d/dx. a(n) = D^(2*n-1)(f(x)) evaluated at x = 0, where f(x) = 1+x+x^2/2! and D is the operator f(x)*d/dx.
Other generating functions: cosh(int {t = 0..x} 1/cos(t)) = 1+x^2/2!+5*x^4/4!+61*x^6/6!+1385*x^8/8!+.... Cf. A012131.
A(x) := arcsinh(tan(x)) = log(sec(x)+tan(x)) = x+x^3/3!+5*x^5/5!+61*x^7/7!+1385*x^9/9!+.... A(x) satisfies A'(x) = cosh(A(x)).
B(x) := Series reversion(log(sec(x)+tan(x))) = x-x^3/3!+5*x^5/5!-61*x^7/7!+1385*x^9/9!-... = arctan(sinh(x)). B(x) satisfies B'(x) = cos(B(x)). (End)
HANKEL transform is A097476. PSUM transform is A173226. - Michael Somos, May 12 2012
a(n+1) - a(n) = A006212(2*n). - Michael Somos, May 12 2012
a(0) = 1 and, for n > 0, a(n) = (-1)^n*((4*n+1)/(2*n+1) - Sum_{k = 1..n} (4^(2*k)/2*k)*C(2*n,2*k-1)*A000367(k)/A002445(k)); see the Bucur et al. link. - L. Edson Jeffery, Sep 17 2012
O.g.f.: Sum_{n>=0} (2*n)!/2^n * x^n / Product_{k=1..n} (1 + k^2*x). - Paul D. Hanna, Sep 20 2012
E.g.f.: 2/Q(0) where Q(k) = 1 + 1/(1 - x^2/(x^2 - 2*(k+1)*(2*k+1)/Q(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 22 2012
G.f.: 1/Q(0) where Q(k) = 1 + x*k*(3*k-1) - x*(k+1)*(2*k+1)*(x*k^2+1)/Q(k+1); (continued fraction, Euler's 1st kind, 3-step). - Sergei N. Gladkovskii, Sep 22 2012
E.g.f.: (2 + x^2 + 2*U(0))/(2 + (2 - x^2)*U(0)) where U(k)= 4*k + 4 + 1/( 1 + x^2/( 2 - x^2 + (2*k+3)*(2*k+4)/U(k+1))) ; (continued fraction, Euler's 1st kind, 3-step). - Sergei N. Gladkovskii, Sep 27 2012
E.g.f.: 1/cos(x)=8*(x^2+1)/(4*x^2 + 8 - x^4*U(0)) where U(k)= 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 - 8*(k+1)*(k+2)*(k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 30 2012
a(n) = sum(k=1..2*n, sum(i=0..k-1, (i-k)^(2*n)*C(2*k,i)*(-1)^(i+k+n)) / (2^(k-1))) for n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012
G.f.: 1/U(0) where U(k)= 1 + x - x*(2*k+1)*(2*k+2)/(1 - x*(2*k+1)*(2*k+2)/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 15 2012
G.f.: 1 + x/G(0) where G(k)= 1 + x - x*(2*k+2)*(2*k+3)/(1 - x*(2*k+2)*(2*k+3)/G(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 16 2012
Let F(x)=sec(x^(1/2))=Sum_{n>=0} a(n)*x^n/(2*n)!, then F(x)=2/(Q(0) + 1) where Q(k)= 1 - x/(2*k+1)/(2*k+2)/(1 - 1/(1 + 1/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
|
|
|
EXAMPLE
|
1 + x + 5*x^2 + 61*x^3 + 1385*x^4 + 50521*x^5 + 2702765*x^6 + 199360981*x^7 + ...
sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + ...
From Gary W. Adamson, Jul 18 2011: (Start)
The first few rows of matrix M are:
1, 1, 0, 0, 0,...
4, 4, 4, 0, 0,...
9, 9, 9, 9, 0,...
16, 16, 16, 16, 16,... (End)
|
|
|
MAPLE
|
series(sec(x), x, 40): SERIESTOSERIESMULT(%): subs(x=sqrt(y), %): seriestolist(%);
A000364_list := proc(n) local S, k, j; S[0] := 1;
for k from 1 to n do S[k] := k*S[k-1] od;
for k from 1 to n do
for j from k to n do
S[j] := (j-k)*S[j-1]+(j-k+1)*S[j] od od;
seq(S[j], j=1..n) end:
A000364_list(16); # Peter Luschny, Apr 02 2012
A000364 := proc(n)
abs(euler(2*n)) ;
end proc: # R. J. Mathar, Mar 14 2013
|
|
|
MATHEMATICA
|
Take[ Range[0, 32]! * CoefficientList[ Series[ Sec[x], {x, 0, 32}], x], {1, 32, 2}] (* from Robert G. Wilson v, Apr 23 2006 *)
Table[Abs[EulerE[2n]], {n, 0, 30}] (* from Ray Chandler, Mar 20 2007 *)
|
|
|
PROG
|
(PARI) {a(n)=local(CF=1+x*O(x^n)); if(n<0, return(0), for(k=1, n, CF=1/(1-(n-k+1)^2*x*CF)); return(Vec(CF)[n+1]))} /* Paul Hanna */
(PARI) {a(n) = if( n<0, 0, (2*n)! * polcoeff( 1 / cos(x + O(x^(2*n + 1))), 2*n))} /* Michael Somos, Jun 18 2002 */
(PARI) {a(n) = local(A); if( n<0, 0, n = 2*n+1 ; A = x * O(x^n); n! * polcoeff( log(1 / cos(x + A) + tan(x + A)), n))} /* Michael Somos, Aug 15 2007 */
(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/2^m * x^m/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)} \\ Paul D. Hanna, Sep 20 2012
(PARI) list(n)=my(v=Vec(1/cos(x+O(x^(2*n+1))))); vector(n, i, v[2*i-1]*(2*i-2)!) \\ Charles R Greathouse IV, Oct 16 2012
(Maxima) a(n):=sum(sum(binomial(k, m)*(-1)^(n+k)/(2^(m-1))*sum(binomial(m, j)*(2*j-m)^(2*n), j, 0, m/2)*(-1)^(k-m), m, 0, k), k, 1, 2*n); [From Kruchinin Vladimir, Aug 05 2010]
(Sage) Algorithm of L. Seidel (1877)
# n -> [a(0), a(1), ..., a(n-1)] for n > 0.
def A000364_list(len) :
R = []; A = {-1:0, 0:1}; k = 0; e = 1
for i in (0..2*len-1) :
Am = 0; A[k + e] = 0; e = -e
for j in (0..i) : Am += A[k]; A[k] = Am; k += e
if e < 0 : R.append(A[-i//2])
return R
A000364_list(17) # Peter Luschny, Mar 31 2012
(Maxima) a[n]:=if n=0 then 1 else sum(sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(i+k+n), i, 0, k-1)/ (2^(k-1)), k, 1, 2*n); makelist(a[n], n, 0, 16); [Vladimir Kruchinin, Oct 05 2012]
|
|
|
CROSSREFS
|
Cf. A000111, A000182, A011248, A060075, A013525, A000816, A002436, A000464, A002105, A002439, A079144, A158690.
Essentially same as A028296 and A122045.
First column of triangle A060074.
Two main diagonals of triangle A060058 (as iterated sums of squares).
Absolute values of row sums of A160485. [Johannes W. Meijer, Jul 06 2009]
Left edge of triangle A210108, see also A125053.
Sequence in context: A096537 A115047 A028296 * A159316 A201254 A116163
Adjacent sequences: A000361 A000362 A000363 * A000365 A000366 A000367
|
|
|
KEYWORD
|
nonn,easy,nice,core
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
