A028296 -id:A028296 - cp's OEIS frontend

A000364 Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).

1, 1, 5, 61, 1385, 50521, 2702765, 199360981, 19391512145, 2404879675441, 370371188237525, 69348874393137901, 15514534163557086905, 4087072509293123892361, 1252259641403629865468285, 441543893249023104553682821, 177519391579539289436664789665

Offset: 0

N. J. A. Sloane

Inverse Gudermannian gd^(-1)(x) = log(sec(x) + tan(x)) = log(tan(Pi/4 + x/2)) = arctanh(sin(x)) = 2 * arctanh(tan(x/2)) = 2 * arctanh(csc(x) - cot(x)). - Michael Somos, Mar 19 2011
a(n) is the number of downup permutations of [2n]. Example: a(2)=5 counts 4231, 4132, 3241, 3142, 2143. - David Callan, Nov 21 2011
a(n) is the number of increasing full binary trees on vertices {0,1,2,...,2n} for which the leftmost leaf is labeled 2n. - David Callan, Nov 21 2011
a(n) is the number of unordered increasing trees of size 2n+1 with only even degrees allowed and degree-weight generating function given by cosh(t). - Markus Kuba, Sep 13 2014
a(n) is the number of standard Young tableaux of skew shape (n+1,n,n-1,...,3,2)/(n-1,n-2,...2,1). - Ran Pan, Apr 10 2015
Since cos(z) has a root at z = Pi/2 and no other root in C with a smaller |z|, the radius of convergence of the e.g.f. (intended complex-valued) is Pi/2 = A019669 (see also A028296). - Stanislav Sykora, Oct 07 2016
All terms are odd. - Alois P. Heinz, Jul 22 2018
The sequence starting with a(1) is periodic modulo any odd prime p. The minimal period is (p-1)/2 if p == 1 mod 4 and p-1 if p == 3 mod 4 [Knuth & Buckholtz, 1967, Theorem 2]. - Allen Stenger, Aug 03 2020
Conjecture: taking the sequence [a(n) : n >= 1] modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, ...] with an apparent period of 6 = phi(21)/2. - Peter Bala, May 08 2023

			G.f. = 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)

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).
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
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.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 420.
G. Chrystal, Algebra, Vol. II, p. 342.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 110.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
L. Euler, Inst. Calc. Diff., Section 224.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
L. Seidel, Über 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.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 5 and 33, pages 41, 314.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 269.

Seiichi Manyama, Table of n, a(n) for n = 0..242 (terms 0..99 from N. J. A. Sloane)
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.-P. Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, Electron. J. Combin., 22 #1 (2015) P1.59; see p. 8.
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
Paul 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 [math.CA], 2009.
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993.
Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and S. T. Chapman, I prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly, Amer. Math. Monthly, 122 (2015), 195-216.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
A. Bucur, J. Lopez-Bonilla, and 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.
Elliot J. Carr and Matthew J. Simpson, Accurate and efficient calculation of response times for groundwater flow, arXiv:1707.06331 [physics.flu-dyn], 2017.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022.
Bishal Deb and Alan D. Sokal, Continued fractions for cycle-alternating permutations, arXiv:2304.06545 [math.CO], 2023.
K. Dilcher and C. Vignat, Euler and the Strong Law of Small Numbers, Amer. Math. Mnthly, 123 (May 2016), 486-490.
Filippo Disanto and Emanuele Munarini, Local height in weighted Dyck models of random walks and the variability of the number of coalescent histories for caterpillar-shaped gene trees and species trees, SN Applied Sciences (2019), 1:578.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
A. L. Edmonds and S, Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
C. J. Fewster and D. Siemssen, Enumerating Permutations by their Run Structure, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144.
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.
Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
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. (Annotated scanned copy)
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, vol.6, no.1, #R21, (1999).
Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688.
D. E. Knuth and Thomas J. Buckholtz Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688. [Annotated scanned copy]
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
Guodong Liu, On Congruences of Euler Numbers Modulo an Odd Square, Fib. Q., 43,2 (2005), 132-136.
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]
F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.
Peter Luschny, Approximation, inclusion and asymptotics of the Euler numbers.
J. Malenfant, Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers, arxiv:1103.1585 [math.NT], 2011.
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.
R. Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
Hisanori Mishima, Factorizations of Euler numbers n=0..78, n=80..106.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Simon Plouffe, 68000 terms, up to E(34000) (2.1 gigas).
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
Paolo Emilio Ricci, Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey, Axioms (2019) Vol. 8, No. 2, 50.
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
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, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
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.
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, arXiv:math/0603520 [math.CO], 2006.
R. P. Stanley, Permutations.
M. A. Stern, Zur Theorie der Eulerschen Zahlen, J. Reine Angew. Math., 79 (1875), 67-98.
Zhi-Wei Sun, On Euler numbers modulo powers of two, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 371-380.
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.

Cf. A000111, A000182, A011248, A019669, A034947, 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, A076552. Cf. A255881.
Bisection (even part) of A317139.
The sequences [(-k^2)^n*Euler(2*n, 1/k), n = 0, 1, ...] are: A000007 (k=1), A000364 (k=2), |A210657| (k=3), A000281 (k=4), A272158 (k=5), A002438 (k=6), A273031 (k=7).

series(sec(x),x,40): SERIESTOSERIESMULT(%): subs(x=sqrt(y),%): seriestolist(%);
# end of program
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

Take[ Range[0, 32]! * CoefficientList[ Series[ Sec[x], {x, 0, 32}], x], {1, 32, 2}] (* Robert G. Wilson v, Apr 23 2006 *)
Table[Abs[EulerE[2n]], {n, 0, 30}] (* Ray Chandler, Mar 20 2007 *)
a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Sec[ x], {x, 0, m}]]]; (* Michael Somos, Nov 22 2013 *)
a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, m! SeriesCoefficient[ InverseGudermannian[ x], {x, 0, m}]]]; (* Michael Somos, Nov 22 2013 *)
a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]* (2j-m)^(2n), {j, 0, m/2}] (-1)^(k-m), {m, 0, k}], {k, 1, 2n}]; Table[ a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 26 2019, after Vladimir Kruchinin *)
a[0] := 1; a[n_] := a[n] = -Sum[a[n - k]/(2 k)!, {k, 1, n}]; Map[(-1)^# (2 #)! a[#] &, Range[0, 16]] (* Oliver Seipel, May 18 2024 *)

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); /* Vladimir Kruchinin, Aug 05 2010 */

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 */

{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 D. Hanna Oct 07 2005

{a(n) = if( n<0, 0, (2*n)! * polcoeff( 1 / cos(x + O(x^(2*n + 1))), 2*n))}; /* Michael Somos, Jun 18 2002 */

{a(n) = my(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 */

{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

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

a(n)=subst(bernpol(2*n+1),'x,1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1) \\ Charles R Greathouse IV, Dec 10 2014

a(n)=abs(eulerfrac(2*n)) \\ Charles R Greathouse IV, Mar 23 2022

\\ Based on an algorithm of Peter Bala, cf. link in A110501.
upto(n) = my(v1, v2, v3); v1 = vector(n+1, i, 0); v1[1] = 1; v2 = vector(n, i, i^2); v3 = v1; for(i=2, n+1, for(j=2, i-1, v1[j] += v2[i-j+1]*v1[j-1]); v1[i] = v1[i-1]; v3[i] = v1[i]); v3 \\ Mikhail Kurkov, Aug 30 2025

from functools import lru_cache
from math import comb
@lru_cache(maxsize=None)
def A000364(n): return 1 if n == 0 else (1 if n % 2 else -1)*sum((-1 if i % 2 else 1)*A000364(i)*comb(2*n,2*i) for i in range(n)) # Chai Wah Wu, Jan 14 2022

# after Mikhail Kurkov, based on an algorithm of Peter Bala, cf. link in A110501.
def euler_list(len: int) -> list[int]:
    if len == 0: return []
    v1 = [1] + [0] * (len - 1)
    v2 = [i**2 for i in range(1, len + 1)]
    result = [0] * len
    result[0] = 1
    for i in range(1, len):
        for j in range(1, i):
            v1[j] += v2[i - j] * v1[j - 1]
        v1[i] = v1[i - 1]
        result[i] = v1[i]
    return result
print(euler_list(1000))  # Peter Luschny, Aug 30 2025

# 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

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]
Also, for positive odd n, log(2) - Sum_{k = 1..(n-1)/2} (-1)^(k-1)/k ~ (-1)^((n-1)/2) * Sum_{k >= 0} (-1)^k*E(k)/n^(2*k+1), where E(k) is the k-th Euler number, by Borwein, Borwein, and Dilcher, Lemma 2 with f(x) := 1/(x + 1/2), h := 1/2 and then replace x with (n-1)/2. - Peter Bala, Oct 29 2016
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 Deléham, 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(j) (mod 2^n) if and only if k == j (mod 2^n) (k and j  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^(2*n+2)*(2*n)!/Pi^(2*n+1) as n -> infinity. [corrected by Vaclav Kotesovec, Jul 10 2021]
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)*binomial(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 due to T. J. Stieltjes). - Paul D. Hanna, Oct 07 2005
a(n) = (Integral_{t=0..Pi} log(tan(t/2)^2)^(2n)dt)/Pi^(2n+1). - Logan Kleinwaks (kleinwaks(AT)alumni.princeton.edu), Mar 15 2007
From Peter Bala, Mar 24 2009: (Start)
Basic hypergeometric generating function: 2*exp(-t)*Sum {n >= 0} 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. - Vladimir Kruchinin, Aug 05 2010
If n is prime, then a(n)==1 (mod 2*n). - Vladimir Shevelev, Sep 04 2010
From Peter Bala, Jan 21 2011: (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)
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)* multinomial(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 examples). - Gary W. Adamson, Jul 18 2011
E.g.f. A'(x) satisfies the differential equation A'(x)=cos(A(x)). - Vladimir Kruchinin, Nov 03 2011
From Peter Bala, Nov 28 2011: (Start)
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(Integral_{t = 0..x} 1/cos(t)) dt = 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)*binomial(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
From Sergei N. Gladkovskii, Oct 31 2011 to Oct 11 2013: (Start)
Continued fractions:
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).
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))).
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).
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))).
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))).
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)).
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)).
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))).
G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/Q(k+1)).
E.g.f.: 1/cos(x) = 1 + x^2/(2-x^2)*Q(0), where Q(k) = 1 - 2*x^2*(k+1)*(2*k+1)/( 2*x^2*(k+1)*(2*k+1)+ (12-x^2 + 14*k + 4*k^2)*(2-x^2 + 6*k + 4*k^2)/Q(k+1)). (End)
a(n) = Sum_{k=1..2*n} (Sum_{i=0..k-1} (i-k)^(2*n)*binomial(2*k,i)*(-1)^(i+k+n)) / 2^(k-1) for n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012
It appears that a(n) = 3*A076552(n -1) + 2*(-1)^n for n >= 1. Conjectural congruences: a(2*n) == 5 (mod 60) for n >= 1 and a(2*n+1) == 1 (mod 60) for n >= 0. - Peter Bala, Jul 26 2013
From Peter Bala, Mar 09 2015: (Start)
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - sqrt(-x)*(2*k + 1)) = Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 + x*(2*k + 1)^2).
O.g.f. is 1 + x*d/dx(log(F(x))), where F(x) = 1 + x + 3*x^2 + 23*x^3 + 371*x^4 + ... is the o.g.f. for A255881. (End)
Sum_(n >= 1, A034947(n)/n^(2d+1)) = a(d)*Pi^(2d+1)/(2^(2d+2)-2)(2d)! for d >= 0; see Allouche and Sondow, 2015. - Jonathan Sondow, Mar 21 2015
Asymptotic expansion: 4*(4*n/(Pi*e))^(2*n+1/2)*exp(1/2+1/(24*n)-1/(2880*n^3) +1/(40320*n^5)-...). (See the Luschny link.) - Peter Luschny, Jul 14 2015
a(n) = 2*(-1)^n*Im(Li_{-2n}(i)), where Li_n(x) is polylogarithm, i=sqrt(-1). - Vladimir Reshetnikov, Oct 22 2015
Limit_{n->infinity} ((2*n)!/a(n))^(1/(2*n)) = Pi/2. - Stanislav Sykora, Oct 07 2016
O.g.f.: 1/(1 + x - 2*x/(1 - 2*x/(1 + x - 12*x/(1 - 12*x/(1 + x - 30*x/(1 - 30*x/(1 + x - ... - (2*n - 1)*(2*n)*x/(1 - (2*n - 1)*(2*n)*x/(1 + x - ... ))))))))). - Peter Bala, Nov 09 2017
For n>0, a(n) = (-PolyGamma(2*n, 1/4) / 2^(2*n - 1) - (2^(2*n + 2) - 2) * Gamma(2*n + 1) * zeta(2*n + 1)) / Pi^(2*n + 1). - Vaclav Kotesovec, May 04 2020
a(n) ~ 2^(4*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)) * exp(Sum_{k>=1} bernoulli(k+1) / (k*(k+1)*2^k*n^k)). - Vaclav Kotesovec, Mar 05 2021
Peter Bala's conjectured congruences, a(2n) == 5 (mod 60) for n >= 1 and a(2n + 1) == 1 (mod 60), hold due to the results of Stern (mod 4) and Knuth & Buckholtz (mod 3 and 5). - Charles R Greathouse IV, Mar 23 2022

Typo in name corrected by Anders Claesson, Dec 01 2015

A122045 Euler (or secant) numbers E(n).

Original entry on oeis.org

1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0, 370371188237525, 0, -69348874393137901, 0, 15514534163557086905, 0, -4087072509293123892361, 0, 1252259641403629865468285, 0, -441543893249023104553682821

Offset: 0

Roger L. Bagula, Sep 13 2006

sign

The convention in the OEIS is that the alternate zeros are normally omitted in such sequences. See A000364 for the official version of this sequence.

Odd primes p such that p | E(p-1) are primes p == 1 (mod 4), A002144. Conjecture: odd composites m such that m | E(m-1) are Carmichael numbers m such that p == 1 (mod 4) for every prime p|m, A265237. - Thomas Ordowski, Feb 06 2020

			G.f. = 1 - x^2 + 5*x^4 - 61*x^6 + 1385*x^8 - 50521*x^10 + 2702765*x^12 + ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 5, page 41.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Barbero G., Juan Margalef-Bentabol, and Eduardo J. S. Villaseñor, A two-sided Faulhaber-like formula involving Bernoulli polynomials, arXiv:2002.00550 [math.NT], 2020.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23.
Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, Finite Sums Involving Fibonacci and Lucas Numbers, J. Int. Seq. (2024). See p. 2.
Michael D. Hirschhorn, Binomial Identities and Congruences for Euler Numbers, Fibonacci Quart. 53 (2015), no. 4, 319-322.
Guodong Liu, Generating functions and generalized Euler numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 2 (2008), 29-34. See p 32.
F. Luca, A. Pizarro-Madariaga, and C. Pomerance, On the counting function of irregular primes, 2014.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
A. Niedermaier and J. Remmel, Analogues of up-down permutations for colored permutations, J. Int. Seq. 13 (2010) 10.5.6.
Bruce E. Sagan, Generalized Euler numbers and ordered set partitions, arXiv:2501.07692 [math.NT], 2025. See p. 2.
T. J. Stieltjes, Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sc. Toulouse 3 (1889) 1-17.

Cf. A000364, A002144, A028296, A265237, A331978, A350972.

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Cosh(x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 13 2020

seq(euler(n) , n=0..31); # Zerinvary Lajos, Mar 15 2009
P := proc(n,x) option remember; if n = 0 then 1 else
   (n*x+(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x); expand(%) fi end:
A122045 := n -> (-1)^n*subs(x=-1, P(n,x)):
seq(A122045(n), n=0..30);  # Peter Luschny, Mar 07 2014
ptan := proc(n) option remember; if irem(n, 2) = 1 then 0 else
    -add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1,2) fi end:
A122045 := n -> ifelse(n = 0, 1, ptan(n)):
seq(A122045(n), n = 0..30); # Peter Luschny, Jun 06 2022

Table[EulerE[n], {n, 0, 30}]
Range[0, 30]! CoefficientList[ Series[ Sech[x], {x, 0, 30}], x] (* Robert G. Wilson v, Aug 08 2018 *)
a[0] := 1; a[n_?OddQ] := 0; a[n_?EvenQ] := a[n] = -Sum[a[k] Binomial[n, k], {k, 0, n - 1, 2}]; Map[a, Range[0, 30]] (* Oliver Seipel, May 20 2024 *)

a[n]:=if n<2 then 1-n else sum(-a[n-2*k]*binomial(n,2*k),k,1,floor(n/2));
makelist(a[n],n,0,50); /* Tani Akinari, Sep 15 2023 */

x='x+O('x^66); Vec(serlaplace(1/cosh(x))) \\ Joerg Arndt, Mar 10 2014

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1,x), x, 3/4) - subst(bernpol(n+1,x), x, 1/4))/(n+1); \\ Michel Marcus, May 20 2017

from sympy import bernoulli as B
def a(n): return int(2**n*2**(n + 1)*(B(n + 1, 3/4) - B(n + 1, 1/4))/(n + 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017, after PARI code by Michel Marcus

from functools import cache
from math import comb as binomial
@cache
def ptan(n):   # see also A331978 and A350972.
    return (0 if n % 2 == 1 else
    -sum(binomial(n,k) * ptan(n-k) if k > 0 else 1 for k in range(0, n-1, 2)))
def A122045(n): return 1 if n == 0 else ptan(n)
print([A122045(n) for n in range(31)])  # Peter Luschny, Jun 06 2022

[euler_number(i) for i in range(31)] # Zerinvary Lajos, Mar 15 2009

E.g.f.: sech(x). - Michael Somos, Mar 11 2014
a(n) = (Sum_{k>=0} (-1)^k*(2*k+1)^n)*2. - Gottfried Helms, Mar 09 2012
From Sergei N. Gladkovskii, Oct 14 2012 - Oct 13 2013: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x + x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: 1/U(0) where U(k) = 1 - x^2 + x^2*(2*k+1)*(2*k+2)/(1 + x^2*(2*k+1)*(2*k+2)/ U(k+1)).
E.g.f.: (1-x)/U(0) where U(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1))).
E.g.f.: 1 - x^2/U(0) where U(k) = (2*k+1)*(2*k+2) + x^2 - x^2*(2*k+1)*(2*k+2)/U(k+1).
E.g.f.: 1/U(0) where U(k) = 1 + x^2/(2*(2*k+1)*(4*k+1) - 2*x^2*(2*k+1)*(4*k+1)/(x^2 + 4*(4*k+3)*(k+1)/U(k+1))).
E.g.f.: (2 + x^4/(U(0)*(x^2-2) - 2))/(2-x^2) where U(k) = 4*k + 4 + 1/(1 + x^2/(2 - x^2 + (2*k+3)*(2*k+4)/U(k+1))).
G.f.: 1/G(0) where G(k) = 1 + x^2*(2*k+1)^2/(1 + x^2*(2*k+2)^2/G(k+1)) (due to T. J. Stieltjes).
G.f.: 1/S(0) where S(k) = 1 + x^2*(k+1)^2/S(k+1) (due to T. J. Stieltjes).
G.f.: 1 - x/(1+x) + x/(1+x)/Q(0) where Q(k) = 1 - x + x*(k+2)/(1 - x*(k+1)/Q(k+1)).
G.f.: -(1/x)/Q(0) where Q(k) = -1/x + (k+1)^2/Q(k+1) (due to T. J. Stieltjes).
G.f.: (1/(1-x))/Q(0) + 1/(1-x) where Q(k) = 1 - 1/x + (k+1)*(k+2)/Q(k+1).
G.f.: (x/(x-1))/Q(0) + 1/(1-x) where Q(k) = 1 - x + x^2*(k+1)*(k+2)/Q(k+1).
G.f.: 1 - x/(1+x) + (x/(1+x))/Q(0) where Q(k) = 1 + x + (k+1)*(k+2)*x^2/Q(k+1).
E.g.f.: 1 - T(0)*x^2/(2+x^2) where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/(x^2*(2*k+1)*(2*k+2) - ((2*k+1)*(2*k+2) + x^2)*((2*k+3)*(2*k+4) + x^2)/T(k+1)).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + 1/T(k+1)). (End)
a(n) = 2^(2*n+1)*(zeta(-n,1/4) - zeta(-n,3/4)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) = 2^n*(2^(n+1)/(n+1))*(B(n+1, 3/4) - B(n+1, 1/4)) where B(n,x) is the n-th Bernoulli polynomial. See Liu link. - Michel Marcus, May 20 2017 [This is the same as: a(n) = -4^(n+1)*B(n+1, 1/4)*((n+1) mod 2)/(n+1). Peter Luschny, Oct 30 2020]
a(n) = 2*Im(PolyLog(-n, I)). - Peter Luschny, Sep 29 2020
a(4n) == 5 (mod 60) and a(4n+2) == -1 (mod 60). See Hirschhorn. - Michel Marcus, Jan 11 2022
For n > 1, a(n) = -Sum_{k=1..floor(n/2)} a(n-2*k)*binomial(n,2*k). - Tani Akinari, Sep 15 2023

Edited by N. J. A. Sloane, Sep 17 2006

A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

Original entry on oeis.org

1, 1, 7, 121, 3907, 202741, 15430207, 1619195761, 224061282907, 39531606447181, 8661323866026007, 2307185279184885001, 734307168916191403507, 275199311597682485597221, 119956934012963778952439407

Offset: 0

Ralf Stephan, Apr 30 2004

With alternating signs, e.g.f.: 1/(2-cos(x)).
7 divides a(3n+2). Ira Gessel remarks: For any odd prime p, the coefficients of 1/(2-cosh(x)) as e.g.f. are periodic with period dividing p-1.
Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
a(n) is the number of ordered set partitions of {1,2,...,2n} into even size blocks. - Geoffrey Critzer, Dec 03 2012
Except for a(0), row sums of A241171. - Peter Bala, Aug 20 2014
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255928. - Peter Bala, Mar 13 2015
Also the 2-packed words of degree n; cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words for 0<=k<=5. - Peter Luschny, Jul 06 2015

Seiichi Manyama, Table of n, a(n) for n = 0..235 (terms 0..40 from Peter Luschny)
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A241171, A210676, A210657, A028296, A210672, A210674, A255928.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];g(1); # N. J. A. Sloane, Mar 28 2012

nn=30;Select[Range[0,nn]!CoefficientList[Series[1/(2-Cosh[x]),{x,0,nn}],x],#>0&]  (* Geoffrey Critzer, Dec 03 2012 *)
a[0]=1; a[n_] := Sum[1/2*(1+(-1)^(2*n))*Sum[((-1)^(k-j)*Binomial[k, j]*Sum[(j-2*i )^(2*n)*Binomial[j, i], {i, 0, j}])/2^j, {j, 1, k}], {k, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)

a(n):=b(2*n+2);
b(n):=sum(((sum(((sum((j-2*i)^n*binomial(j,i),i,0,j))*(-1)^(k-j)*binomial(k,j))/2^(j),j,1,k))*((-1)^n+1))/2,k,1,n/2); /* Vladimir Kruchinin, Apr 23 2011 */

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(i),i,0,k-1)/(2^(k-1)),k,1,2*n); /* Vladimir Kruchinin, Oct 05 2012 */

a(n) = if (n == 0, 1, sum(k=1, n, binomial(2*n, 2*n-2*k)*a(n-k)));

def A094088(n) :
    @CachedFunction
    def intern(n) :
        if n == 0 : return 1
        if n % 2 != 0 : return 0
        return add(intern(k)*binomial(n,k) for k in range(n)[::2])
    return intern(2*n)
[A094088(n) for n in (0..14)]  # Peter Luschny, Jul 14 2012

1/(2-cosh(x)) = Sum_{n>=0} a(n)x^(2n)/(2n)! = 1 + x^2/2 + 7x^4/24 + 121x^6/720 + ...
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} C(2n, 2n-2k)*a(n-k).
a(0)=1 and, for n>=1, a(n)=b(2*n) where b(n) = sum(k=1..n/2,((sum(j=1..k, ((sum(i=0..j,(j-2*i)^n*binomial(j,i)))*(-1)^(k-j)*binomial(k,j))/2^(j)))*((-1)^n+1))/2). - Vladimir Kruchinin, Apr 23 2011
E.g.f.: 1/(2-cosh(x))=8*(1-x^2)/(8 - 12*x^2 + 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) * binomial(2*k,i) * (-1)^i )/2^(k-1), n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012
a(n) ~ 2*(2*n)! /(sqrt(3) * (log(2+sqrt(3)))^(2*n+1)). - Vaclav Kotesovec, Oct 19 2013

Corrected definition, Joerg Arndt, Apr 26 2011

A065547 Triangle of Salie numbers.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -17, 17, -6, 1, 0, 155, -155, 55, -10, 1, 0, -2073, 2073, -736, 135, -15, 1, 0, 38227, -38227, 13573, -2492, 280, -21, 1, 0, -929569, 929569, -330058, 60605, -6818, 518, -28, 1, 0, 28820619, -28820619, 10233219, -1879038, 211419, -16086, 882, -36, 1, 0, -1109652905

Offset: 0

Wouter Meeussen, Dec 02 2001

Coefficients of polynomials H(n,x) related to Euler polynomials through H(n,x(x-1)) = E(2n,x).

			Triangle begins:
 1;
 0,   1;
 0,  -1,    1;
 0,   3,   -3,  1;
 0, -17,   17, -6,   1;
 0, 155, -155, 55, -10, 1;
 ...

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
Ira M. Gessel and X. G. Viennot, Determinants, paths and plane partitions, 1989, p. 27, eqn 12.1.
A. F. Horadam, Generation of Genocchi polynomials of first order by recurrence relation, Fib. Quart. 2 (1992), 239-243.

Sum_{k>=0} (-1)^(n+k)*2^(n-k)*T(n, k) = A005647(n). Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*T(n, k) = A000795(n). Sum_{k>=0} (-1)^(n+k)*T(n, k) = A006846(n), where A006846 = Hammersley's polynomial p_n(1). - Philippe Deléham, Feb 26 2004.
Column sequences (without leading zeros) give, for k=1..10: A065547 (twice), A095652-9.
Cf. A000795, A005647, A000035.
See A085707 for unsigned and transposed version.
See A098435 for negative values of n, k.

h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x - 1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 02 2013 *)

{ S2(n, k) = (1/k!)*sum(i=0,k,(-1)^(k-i)*binomial(k,i)*i^n) }{ Eu(n) = sum(m=0,n,(-1)^m*m!*S2(n+1,m+1)*(-1)^floor(m/4)*2^-floor(m/2)*((m+1)%4!=0)) } T(n,k)=if(nRalf Stephan

E.g.f.: Sum_{n, k=0..oo} T(n, k) t^k x^(2n)/(2n)! = cosh(sqrt(1+4t) x/2) / cosh(x/2).

T(k, n) = Sum_{i=0..n-k} A028296(i)/4^(n-k)*C(2n, 2i)*C(n-i, n-k-i), or 0 if n

Polynomial recurrences: x^n = Sum_{0<=2i<=n} C(n, 2i)*H(n-i, x); (1/4+x)^n = Sum_{m=0..n} C(2n, 2m)*(1/4)^(n-m)*H(m, x).

Dumont/Zeng give a continued fraction and other formulas.

Triangle T(n, k) read by rows; given by [0, -1, -2, -4, -6, -9, -12, -16, ...] DELTA A000035, where DELTA is Deléham's operator defined in A084938.

Sum_{k=0..n} (-4)^(n-k)*T(n,k) = A000364(n) (Euler numbers). - Philippe Deléham, Oct 25 2006

Edited by Ralf Stephan, Sep 08 2004

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -2, 22, -602, 30742, -2523002, 303692662, -50402079002, 11030684333782, -3077986048956602, 1066578948824962102, -449342758735568563802, 226182806795367665865622, -134065091768709178087428602, 92423044260377387363207812342, -73323347841467639992211297199002

Offset: 0

N. J. A. Sloane, Mar 28 2012

sign

The version without signs has an interpretation as a sum over marked Schröder paths. See the Josuat-Verges and Kim reference.
Consider the sequence defined by a(0)=1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Apparently a(n) = 2*(-1)^n*A002114(n). - R. J. Mathar, Mar 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.
Zhi-Hong Sun, On the further properties of U_n, arXiv:1203.5977 [math.NT], 2012.

Cf. A002114, A028296, A094088, A210657, A210672, A210674, A210676, A249939, A255882, A241171.

A210657:=proc(n) option remember;
   if n=0 then 1
   else -2*add(binomial(2*n,2*k)*procname(n-k),k=1..floor(n)); fi;
end;
[seq(f(n),n=0..20)];
# Second program:
a := (n) -> 2*36^n*(Zeta(0,-n*2,1/6)-Zeta(0,-n*2,2/3)):
seq(a(n), n=0..15); # Peter Luschny, Mar 11 2015

nmax=20; Table[(CoefficientList[Series[1/(2*Cosh[x]-1), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)
Table[9^n EulerE[2 n, 1/3], {n, 0, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

a(n)=polcoeff(sum(m=0, n, (2*m)!*(-x)^m/prod(k=1, m, 1-k^2*x +x*O(x^n)) ), n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 17 2012

O.g.f.: Sum_{n>=0} (2*n)! * (-x)^n / Product_{k=1..n} (1 - k^2*x). - Paul D. Hanna, Sep 17 2012
E.g.f.: 1/(2*cosh(x) - 1) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. - Paul D. Hanna, Oct 30 2014
E.g.f.: cos(z/2)/cos(3z/2) = Sum_{n>=0} abs(a(n))*x^(2*n)/(2*n)!. - Olivier Gérard, Feb 12 2014
From Peter Bala, Mar 09 2015: (Start)
a(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial.
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(3*k + 1)^2).
O.g.f. as a continued fraction: 1/(1 + (3^2 - 1^2)*x/(4 + 12^2*x/(4 + (18^2 - 2^2)*x/(4 + 24^2*x/(4 + (30^2 - 2^2)*x/(4 + 36^2*x/(4 + ... ))))))) = 1 - 2*x + 22*x^2 - 602*x^3 + 30742*x^4 - .... See Josuat-Vergès and Kim, p. 23.
The expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) appears to have integer coefficients. See A255882. (End)
a(n) = 2*36^n*(zeta(-n*2,1/6)-zeta(-n*2,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) ~ 2 * (-1)^n * (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k=0..n} A241171(n, k)*(-2)^k. - Peter Luschny, Sep 03 2022

A249939 E.g.f.: 1/(5 - 4*cosh(x)).

Original entry on oeis.org

1, 4, 100, 6244, 727780, 136330084, 37455423460, 14188457293924, 7087539575975140, 4514046217675793764, 3570250394992512270820, 3433125893070920512725604, 3944372161432193963534198500, 5336301013125557989981503385444, 8396749419933421378024498580446180

Offset: 0

Paul D. Hanna, Nov 19 2014

nonn

a(n) = 4*A242858(2*n) for n>0.
a(n) = A249940(n)/3.
a(n) == 4 (mod 96) for n>0.

			E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
where 3*A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 + 2*exp(49*x)/2^7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A249938, A249940, A247082, A250914, A250915, A242858.

Cf. A210676, A210657, A028296, A094088, A210672, A210674.

/* E.g.f.: 1/(5 - 4*cosh(x)) */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, 2, 26, 842, 50906, 4946282, 704888186, 138502957322, 35887046307866, 11855682722913962, 4863821092813045946, 2425978759725443056202, 1445750991051368583278426, 1014551931766896667943384042, 828063237870027116855857421306, 777768202388460616924079724057482

Offset: 0

N. J. A. Sloane, Mar 28 2012

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255929. - Peter Bala, Mar 13 2015
The Stirling-Bernoulli transform of Fibonacci(n+1) = 1, 1, 2, 3, 5, 8, 13, ... is 1, 0, 2, 0, 26, 0, 842, 0, 50906, 0, ... - Philippe Deléham, May 25 2015

G. C. Greubel, Table of n, a(n) for n = 0..220
Hacène Belbachir, Yahia Djemmada, On central Fubini-like numbers and polynomials, arXiv:1811.06734 [math.CO], 2018. See p. 4.

Cf. A028296, A094088, A210657, A210674, A210676, A255929, A241171.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(2);

nmax=20; Table[(CoefficientList[Series[1/(3-2*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

a(n) ~ 2*sqrt(Pi/5) * n^(2*n+1/2) / (exp(2*n) * (log((1+sqrt(5))/2))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
E.g.f.: 1/(3-2*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k = 0..2*n} A163626(2*n,k)*A000045(n+1). - Philippe Deléham, May 25 2015
a(n) = Sum_{k=0..n} A241171(n, k)*2^k. - Peter Luschny, Sep 03 2022

A210676 a(0)=1; thereafter a(n) = -3Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -3, 51, -2163, 171231, -21785223, 4065116811, -1045879150683, 354837765112791, -153492920593758543, 82453488412268175171, -53850296379425229208803, 42020794900180632536559951, -38611325264740403135096141463, 41264215393801752999038147563131, -50749285521783354479522581233836523

Offset: 0

N. J. A. Sloane, Mar 28 2012

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255926. - Peter Bala, Mar 13 2015
In general, for c<>0 is e.g.f. = 1/(c+1-c*cosh(x)) (even coefficients). For c > 0 is a(n) ~ 2 * (2*n)! / (sqrt(2*c+1) * (arccosh((c+1)/c))^(2*n+1)). For c < 0 is a(n) ~ (-1)^n * (2*n)! / (sqrt(-2*c-1) * 2^(2*n) * arccos(sqrt((2*c + 1) / (2*c)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A028296, A094088, A210657, A210672, A210674, A249939, A255926.

f:=proc(n,k) option remember;  local i;
if n=0 then 1
else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
g:=k->[seq(f(n,k),n=0..40)];
g(-3);

nmax=20; Table[(CoefficientList[Series[1/(3*Cosh[x]-2), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

E.g.f.: 1/(3*cosh(x)-2) (even coefficients). - Vaclav Kotesovec, Mar 14 2015

a(n) ~ (-1)^n * (2*n)! / (sqrt(5) * 2^(2*n) * (arccos(sqrt(5/6)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

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cp's OEIS Frontend

A034971 a(n) = floor(E_(n+1)/E_(n)) where E_n is n-th Euler number (see A028296 and A000364).

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A239322 Interleave (-1)^n*(A000182(n+1) - A000364(n)), -A028296(n+1).

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A000364 Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).

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A122045 Euler (or secant) numbers E(n).

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A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

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A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

A210672 a(0)=1; thereafter a(n) = 2Sum_{k=1..n} binomial(2n,2k)a(n-k).

A210676 a(0)=1; thereafter a(n) = -3Sum_{k=1..n} binomial(2n,2k)a(n-k).