A002430 -id:A002430 - cp's OEIS frontend

A092505 a(n) = A002430(n) / A046990(n).

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4

Offset: 1

Ralf Stephan, Apr 05 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A002430, A046990, A130654.

[Numerator((-1)^(n - 1)*2^(2*n)*(2^(2*n) - 1)*Bernoulli(2*n) / Factorial(2*n)) / (Numerator(((-4)^n-(-16)^n) * Bernoulli(2*n) / 2 / n / Factorial(2*n))): n in [1..100]]; // Vincenzo Librandi, Jan 13 2019

a(n)=if(n<1,0,numerator(polcoeff(Ser(tan(x)),2*n-1))/numerator(polcoeff(Ser(log(1/cos(x))),2*n)))

\\ Quite wasteful, especially as there is the same bernfrac(2*n) in both. Should reduce to a much simpler form?
A002430(n) = numerator(((-1)^(n-1)) * 2^(2*n) * (2^(2*n)-1)*bernfrac(2*n)/((2*n)!)); \\ After Johannes W. Meijer's May 24 2009 formula in A002430.
A046990(n) = numerator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!); \\ From A046990
A092505(n) = (A002430(n) / A046990(n)); \\ Antti Karttunen, Jan 12 2019

A007814(a(n)) = A130654(n). - Antti Karttunen, Jan 12 2019

A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Alexander Adamchuk, Jun 20 2007, Jun 23 2007

nonn

Conjecture: A092505(n) is always a power of 2. a(n) = Log[ 2, A092505(n) ]. Note that a(n) = 0 iff n is a power of 2; or A002430(2^n) = A046990(2^n) and A092505(2^n) = 1. It appears that a(2k+1) = 1 for k>0. Note that least index k such that a(k) = n is {1, 3, 14, 60, ...} which apparently coincides with A006502(n) = {1, 3, 14, 60, 279, 1251, ...} Related to Fibonacci numbers (see Carlitz reference).
Least index k such that a(k) = n is listed in A131262(n) = {1, 3, 14, 60, 248, ...}. Conjecture: A131262(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n). If this conjecture is true then a(1008) = 5 and a(n)<5 for all n<1008.
Positions of records indeed continue as 1, 3, 14, 60, 248, 1008, 4064, 16320, ..., strongly suggesting union of {1} and A171499. - Antti Karttunen, Jan 13 2019

			A092505(n) begins {1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, ...}.
Thus a(1) = Log[2,1] = 0, a(2) = Log[2,1] = 0, a(3) = Log[2,2] = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16385
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A006502 = Related to Fibonacci numbers.
Cf. A131262 = Least index k such that A130654(k) = n. Cf. A062354 = Sigma(n)*EulerPhi(n).
Cf. also A171499.

a=Series[ Tan[x], {x,0,256} ]; b=Series[ Log[ 1/Cos[x] ], {x,0,256}]; Table[ Log[ 2, Numerator[ SeriesCoefficient[ a, 2n-1 ] ] / Numerator[ SeriesCoefficient[ b, 2n ] ] ], {n,1,128} ]

a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ] = A007814(A092505(n)).

A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

Original entry on oeis.org

1, 1, 17, 929569, 129848163681107301953, 7724760729208487305545342963324697288405380586579904269441, 357302767470032900576643605538835088084055212588960920085261795996340330997333306469144562500392344758421560010463942134842407723273904635849262137252097

Offset: 1

Alexander Adamchuk, Jun 20 2007

nonn

Odd terms in A002430(n) correspond to the indices that are the powers of 2.

			tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = 1*x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^10).
A002430(n) begins {1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 113927491862, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, ...}.
Thus a(1) = 1, a(2) = 1, a(3) = 17, a(4) = 929569, a(5) = 129848163681107301953.

Eric Weisstein's World of Mathematics, Tangent.

Cf. A002430 = Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x). Cf. A001469, A002425, A046990, A089171, A110501, A036968.

Table[ Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ], {n,1,8} ]

a(n) = Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ]. a(n) = A002430(2^(n-1)).

A000182 Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).

Original entry on oeis.org

1, 2, 16, 272, 7936, 353792, 22368256, 1903757312, 209865342976, 29088885112832, 4951498053124096, 1015423886506852352, 246921480190207983616, 70251601603943959887872, 23119184187809597841473536, 8713962757125169296170811392, 3729407703720529571097509625856

Offset: 1

N. J. A. Sloane

Number of Joyce trees with 2n-1 nodes. Number of tremolo permutations of {0,1,...,2n}. - Ralf Stephan, Mar 28 2003
The Hankel transform of this sequence is A000178(n) for n odd = 1, 12, 34560, ...; example: det([1, 2, 16; 2, 16, 272, 16, 272, 7936]) = 34560. - Philippe Deléham, Mar 07 2004
a(n) is the number of increasing labeled full binary trees with 2n-1 vertices. Full binary means every non-leaf vertex has two children, distinguished as left and right; labeled means the vertices are labeled 1,2,...,2n-1; increasing means every child has a label greater than its parent. - David Callan, Nov 29 2007
From Micha Hofri (hofri(AT)wpi.edu), May 27 2009: (Start)
a(n) was found to be the number of permutations of [2n] which when inserted in order, to form a binary search tree, yield the maximally full possible tree (with only one single-child node).
The e.g.f. is sec^2(x)=1+tan^2(x), and the same coefficients can be manufactured from the tan(x) itself, which is the e.g.f. for the number of trees as above for odd number of nodes. (End)
a(n) is the number of increasing strict binary trees with 2n-1 nodes. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 07 2014
For relations to alternating permutations, Euler and Bernoulli polynomials, zigzag numbers, trigonometric functions, Fourier transform of a square wave, quantum algebras, and integrals over and in n-dimensional hypercubes and over Green functions, see Hodges and Sukumar. For further discussion on the quantum algebra, see the later Hodges and Sukumar reference and the paper by Hetyei presenting connections to the general combinatorial theory of Viennot on orthogonal polynomials, inverse polynomials, tridiagonal matrices, and lattice paths (thereby related to continued fractions and cumulants). - Tom Copeland, Nov 30 2014
The Zigzag Hankel transform is A000178. That is, A000178(2*n - k) = det( [a(i+j - k)]{i,j = 1..n} ) for n>0 and k=0,1. - _Michael Somos, Mar 12 2015
a(n) is the number of standard Young tableaux of skew shape (n,n,n-1,n-2,...,3,2)/(n-1,n-2,n-3,...,2,1). - Ran Pan, Apr 10 2015
For relations to the Sheffer Appell operator calculus and a Riccati differential equation for generating the Meixner-Pollaczek and Krawtchouk orthogonal polynomials, see page 45 of the Feinsilver link and Rzadkowski. - Tom Copeland, Sep 28 2015
For relations to an elliptic curve, a Weierstrass elliptic function, the Lorentz formal group law, a Lie infinitesimal generator, and the Eulerian numbers A008292, see A155585. - Tom Copeland, Sep 30 2015
Absolute values of the alternating sums of the odd-numbered rows (where the single 1 at the apex of the triangle is counted as row #1) of the Eulerian triangle, A008292. The actual alternating sums alternate in sign, e.g., 1, -2, 16, -272, etc. (Even-numbered rows have alternating sums always 0.) - Gregory Gerard Wojnar, Sep 28 2018
The sequence 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 1]. - Allen Stenger, Aug 03 2020
From Peter Bala, Dec 24 2021: (Start)
Conjectures:
1) The sequence taken modulo any integer k eventually becomes periodic with period dividing phi(k).
2) The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k, except when p = 2, n = 1 and k = 1 or 2.
3) For i >= 1 define a_i(n) = a(n+i). The Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i >= 1 the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients. For an example, see A262145.(End)

			tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ... = x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^11).
tanh(x) = x - 1/3*x^3 + 2/15*x^5 - 17/315*x^7 + 62/2835*x^9 - 1382/155925*x^11 + ...
(sec x)^2 = 1 + x^2 + 2/3*x^4 + 17/45*x^6 + ...
a(3)=16 because we have: {1, 3, 2, 5, 4}, {1, 4, 2, 5, 3}, {1, 4, 3, 5, 2},
  {1, 5, 2, 4, 3}, {1, 5, 3, 4, 2}, {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3},
  {2, 4, 3, 5, 1}, {2, 5, 1, 4, 3}, {2, 5, 3, 4, 1}, {3, 4, 1, 5, 2},
  {3, 4, 2, 5, 1}, {3, 5, 1, 4, 2}, {3, 5, 2, 4, 1}, {4, 5, 1, 3, 2},
  {4, 5, 2, 3, 1}. - _Geoffrey Critzer_, May 19 2013

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Index entries for "core" sequences
Index entries for sequences related to boustrophedon transform
Index entries for sequences related to Bernoulli numbers.

A350972 is essentially the same sequence.
a(n)=2^(n-1)*A002105(n). Apart from signs, 2^(2n-2)*A001469(n) = n*a(n).
Cf. A001469, A002430, A036279, A000364 (secant numbers), A000111 (secant-tangent numbers), A024283, A009764. First diagonal of A059419 and of A064190.
Cf. A009006, A009725, A029584, A012509, A009123, A009567.
Equals A002425(n) * 2^A101921(n).
Equals leftmost column of A162005. - Johannes W. Meijer, Jun 27 2009
Cf. A258880, A258901. Cf. A002105, A080635, A234797.
Cf. A019538, A110501.

series(tan(x),x,40);
with(numtheory): a := n-> abs(2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n));
A000182_list := proc(n) local T,k,j; T[1] := 1;
for k from 2 to n do T[k] := (k-1)*T[k-1] od;
   for k from 2 to n do
       for j from k to n do
           T[j] := (j-k)*T[j-1]+(j-k+2)*T[j] od od;
seq(T[j], j=1..n)  end:
A000182_list(15);  # Peter Luschny, Apr 02 2012

Table[ Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}], {n, 0, 7}] (* Victor Adamchik, Oct 05 2005 *)
v[1] = 2; v[n_] /; n >= 2 := v[n] = Sum[ Binomial[2 n - 3, 2 k - 2] v[k] v[n - k], {k, n - 1}]; Table[ v[n]/2, {n, 15}] (* Zerinvary Lajos, Jul 08 2009 *)
Rest@ Union[ Range[0, 29]! CoefficientList[ Series[ Tan[x], {x, 0, 30}], x]] (* Harvey P. Dale, Oct 19 2011; modified by Robert G. Wilson v, Apr 02 2012 *)
t[1, 1] = 1; t[1, 0] = 0; t[n_ /; n > 1, m_] := t[n, m] = m*(m+1)*Sum[t[n-1, k], {k, m-1, n-1}]; a[n_] := t[n, 1]; Table[a[n], {n, 1, 15}]  (* Jean-François Alcover, Jan 02 2013, after A064190 *)
a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m! SeriesCoefficient[ Tan[x], {x, 0, m}]]]; (* Michael Somos, Mar 12 2015 *)
a[ n_] := If[ n < 1, 0, ((-16)^n - (-4)^n) Zeta[1 - 2 n]]; (* Michael Somos, Mar 12 2015 *)
Table[2 PolyGamma[2n - 1, 1/2]/Pi^(2n), {n, 1, 10}] (* Vladimir Reshetnikov, Oct 18 2015 *)
a[ n_] := a[n] = If[ n < 2, Boole[n == 1], Sum[Binomial[2 n - 2, 2 k - 1] a[k] a[n - k], {k, n - 1}]]; (* Michael Somos, Aug 02 2018 *)
a[n_] := (2^(2*n)*(2^(2*n) - 1)*Abs[BernoulliB[2*n]])/(2*n); a /@  Range[20]  (* Stan Wagon, Nov 21 2022 *)

a(n):=sum(sum(binomial(k,r)*sum(sum(binomial(l,j)/2^(j-1)*sum((-1)^(n)*binomial(j,i)*(j-2*i)^(2*n),i,0,floor((j-1)/2))*(-1)^(l-j),j,1,l)*(-1)^l*binomial(r+l-1,r-1),l,1,2*n)*(-1)^(1-r),r,1,k)/k,k,1,2*n); /* Vladimir Kruchinin, Aug 23 2010 */

a[n]:=if n=1 then 1 else 2*sum(sum(binomial(2*j,j+k)*(-4*k^2)^(n-1)*(-1)^k/(4^j),k,1,j),j,1,n-1);
makelist(a[n],n,1,30); /* Tani Akinari, Sep 20 2023 */

{a(n) = if( n<1, 0, ((-4)^n - (-16)^n) * bernfrac(2*n) / (2*n))};

{a(n) = my(an); if( n<2, n==1, an = vector(n, m, 1); for( m=2, n, an[m] = sum( k=1, m-1, binomial(2*m - 2, 2*k - 1) * an[k] * an[m-k])); an[n])}; /* Michael Somos */

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

{a(n) = my(X=x+x*O(x^n),Egf); Egf = x*sum(m=0,n, prod(k=1,m, tanh(2*k*X))); (n-1)!*polcoeff(Egf,n)} /* Paul D. Hanna, May 11 2010 */

/* Continued Fraction for the e.g.f. tan(x), from Paul D. Hanna: */
{a(n)=local(CF=1+O(x)); for(i=1, n, CF=1/(2*(n-i+1)-1-x^2*CF)); (2*n-1)!*polcoeff(x*CF, 2*n-1)}

/* O.g.f. Sum_{n>=1} a(n)*x^n, from Paul D. Hanna Feb 05 2013: */
{a(n)=polcoeff( x+2*x*sum(m=1, n, x^m*prod(k=1, m, (2*k-1)^2/(1+(2*k-1)^2*x +x*O(x^n))) ), n)}

# The objective of this implementation is efficiency.
# n -> [0, a(1), a(2), ..., a(n)] for n > 0.
def A000182_list(n):
    T = [0 for i in range(1, n+2)]
    T[1] = 1
    for k in range(2, n+1):
        T[k] = (k-1)*T[k-1]
    for k in range(2, n+1):
        for j in range(k, n+1):
            T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    return T
print(A000182_list(100)) # Peter Luschny, Aug 07 2011

from sympy import bernoulli
def A000182(n): return abs(((2-(2<<(m:=n<<1)))*bernoulli(m)<Chai Wah Wu, Apr 14 2023

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A000182_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
A000182_list(15) # Peter Luschny, Mar 31 2012

E.g.f.: log(sec x) = Sum_{n > 0} a(n)*x^(2*n)/(2*n)!.
E.g.f.: tan x = Sum_{n >= 0} a(n+1)*x^(2*n+1)/(2*n+1)!.
E.g.f.: (sec x)^2 = Sum_{n >= 0} a(n+1)*x^(2*n)/(2*n)!.
2/(exp(2x)+1) = 1 + Sum_{n>=1} (-1)^(n+1) a(n) x^(2n-1)/(2n-1)! = 1 - x + x^3/3 - 2*x^5/15 + 17*x^7/315 - 62*x^9/2835 + ...
a(n) = 2^(2*n) (2^(2*n) - 1) |B_(2*n)| / (2*n) where B_n are the Bernoulli numbers (A000367/A002445 or A027641/A027642).
Asymptotics: a(n) ~ 2^(2*n+1)*(2*n-1)!/Pi^(2*n).
Sum[2^(2*n + 1 - k)*(-1)^(n + k + 1)*k!*StirlingS2[2*n + 1, k], {k, 1, 2*n + 1}]. - Victor Adamchik, Oct 05 2005
a(n) = abs[c(2*n-1)] where c(n)= 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1) = 2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n) = [ -(1+EN(.))]^n = 2^n * GN(n+1)/(n+1) = 2^n * EP(n,0) = (-1)^n * E(n,-1) = (-2)^n * n! * Lag[n,-P(.,-1)/2] umbrally = (-2)^n * n! * C{T[.,P(.,-1)/2] + n, n} umbrally for the signed Euler numbers EN(n), the Bernoulli numbers Ber(n), the Genocchi numbers GN(n), the Euler polynomials EP(n,t), the Eulerian polynomials E(n,t), the Touchard / Bell polynomials T(n,t), the binomial function C(x,y) = x!/[(x-y)!*y! ] and the polynomials P(j,t) of A131758. - Tom Copeland, Oct 05 2007
a(1) = A094665(0,0)*A156919(0,0) and a(n) = Sum_{k=1..n-1} 2^(n-k-1)*A094665(n-1, k)*A156919(k,0) for n = 2, 3, .., see A162005. - Johannes W. Meijer, Jun 27 2009
G.f.: 1/(1-1*2*x/(1-2*3*x/(1-3*4*x/(1-4*5*x/(1-5*6*x/(1-... (continued fraction). - Paul Barry, Feb 24 2010
From Paul Barry, Mar 29 2010: (Start)
G.f.: 1/(1-2x-12x^2/(1-18x-240x^2/(1-50x-1260x^2/(1-98x-4032x^2/(1-162x-9900x^2/(1-... (continued fraction);
coefficient sequences given by 4*(n+1)^2*(2n+1)*(2n+3) and 2(2n+1)^2 (see Van Fossen Conrad reference). (End)
E.g.f.: x*Sum_{n>=0} Product_{k=1..n} tanh(2*k*x) = Sum_{n>=1} a(n)*x^n/(n-1)!. - Paul D. Hanna, May 11 2010 [corrected by Paul D. Hanna, Sep 28 2023]
a(n) = (-1)^(n+1)*Sum_{j=1..2*n+1} j!*Stirling2(2*n+1,j)*2^(2*n+1-j)*(-1)^j for n >= 0. Vladimir Kruchinin, Aug 23 2010: (Start)
If n is odd such that 2*n-1 is prime, then a(n) == 1 (mod (2*n-1)); if n is even such that 2*n-1 is prime, then a(n) == -1 (mod (2*n-1)). - Vladimir Shevelev, Sep 01 2010
Recursion: a(n) = (-1)^(n-1) + Sum_{i=1..n-1} (-1)^(n-i+1)*C(2*n-1,2*i-1)* a(i). - Vladimir Shevelev, Aug 08 2011
E.g.f.: tan(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! = x/(1 - x^2/(3 - x^2/(5 - x^2/(7 - x^2/(9 - x^2/(11 - x^2/(13 -...))))))) (continued fraction from J. H. Lambert - 1761). - Paul D. Hanna, Sep 21 2011
From Sergei N. Gladkovskii, Oct 31 2011 to Oct 09 2013: (Start)
Continued fractions:
E.g.f.: (sec(x))^2 = 1+x^2/(x^2+U(0)) where U(k) = (k+1)*(2k+1) - 2x^2 + 2x^2*(k+1)*(2k+1)/U(k+1).
E.g.f.: tan(x) = x*T(0) where T(k) = 1-x^2/(x^2-(2k+1)*(2k+3)/T(k+1)).
E.g.f.: tan(x) = x/(G(0)+x) where G(k) = 2*k+1 - 2*x + x/(1 + x/G(k+1)).
E.g.f.: tanh(x) = x/(G(0)-x) where G(k) = k+1 + 2*x - 2*x*(k+1)/G(k+1).
E.g.f.: tan(x) = 2*x - x/W(0) where W(k) = 1 + x^2*(4*k+5)/((4*k+1)*(4*k+3)*(4*k+5) - 4*x^2*(4*k+3) + x^2*(4*k+1)/W(k+1)).
E.g.f.: tan(x) = x/T(0) where T(k) = 1 - 4*k^2 + x^2*(1 - 4*k^2)/T(k+1).
E.g.f.: tan(x) = -3*x/(T(0)+3*x^2) where T(k)= 64*k^3 + 48*k^2 - 4*k*(2*x^2 + 1) - 2*x^2 - 3 - x^4*(4*k -1)*(4*k+7)/T(k+1).
G.f.: 1/G(0) where G(k) = 1 - 2*x*(2*k+1)^2 - x^2*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1).
G.f.: 2*Q(0) - 1 where Q(k) = 1 + x^2*(4*k + 1)^2/(x + x^2*(4*k + 1)^2 - x^2*(4*k + 3)^2*(x + x^2*(4*k + 1)^2)/(x^2*(4*k + 3)^2 + (x + x^2*(4*k + 3)^2)/Q(k+1) )).
G.f.: (1 - 1/G(0))*sqrt(-x), where G(k) = 1 + sqrt(-x) - x*(k+1)^2/G(k+1).
G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/Q(k+1)). (End)
O.g.f.: x + 2*x*Sum_{n>=1} x^n * Product_{k=1..n} (2*k-1)^2 / (1 + (2*k-1)^2*x). - Paul D. Hanna, Feb 05 2013
a(n) = (-4)^n*Li_{1-2*n}(-1). - Peter Luschny, Jun 28 2012
a(n) = (-4)^n*(4^n-1)*Zeta(1-2*n). - Jean-François Alcover, Dec 05 2013
Asymptotic expansion: 4*((2*(2*n-1))/(Pi*e))^(2*n-1/2)*exp(1/2+1/(12*(2*n-1))-1/(360*(2*n-1)^3)+1/(1260*(2*n-1)^5)-...). (See Luschny link.) - Peter Luschny, Jul 14 2015
From Peter Bala, Sep 11 2015: (Start)
The e.g.f. A(x) = tan(x) satisfies the differential equation A''(x) = 2*A(x)*A'(x) with A(0) = 0 and A'(0) = 1, leading to the recurrence a(0) = 0, a(1) = 1, else a(n) = 2*Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i) for the aerated sequence [0, 1, 0, 2, 0, 16, 0, 272, ...].
Note, the same recurrence, but with the initial conditions a(0) = 1 and a(1) = 1, produces the sequence n! and with a(0) = 1/2 and a(1) = 1 produces A080635. Cf. A002105, A234797. (End)
a(n) = 2*polygamma(2*n-1, 1/2)/Pi^(2*n). - Vladimir Reshetnikov, Oct 18 2015
a(n) = 2^(2n-2)*|p(2n-1,-1/2)|, where p_n(x) are the shifted row polynomials of A019538. E.g., a(2) = 2 = 2^2 * |1 + 6(-1/2) + 6(-1/2)^2|. - Tom Copeland, Oct 19 2016
From Peter Bala, May 05 2017: (Start)
With offset 0, the o.g.f. A(x) = 1 + 2*x + 16*x^2 + 272*x^3 + ... has the property that its 4th binomial transform 1/(1 - 4*x) A(x/(1 - 4*x)) has the S-fraction representation 1/(1 - 6*x/(1 - 2*x/(1 - 20*x/(1 - 12*x/(1 - 42*x/(1 - 30*x/(1 - ...))))))), where the coefficients [6, 2, 20, 12, 42, 30, ...] in the partial numerators of the continued fraction are obtained from the sequence [2, 6, 12, 20, ..., n*(n + 1), ...] by swapping adjacent terms. Compare with the S-fraction associated with A(x) given above by Paul Barry.
A(x) = 1/(1 + x - 3*x/(1 - 4*x/(1 + x - 15*x/(1 - 16*x/(1 + x - 35*x/(1 - 36*x/(1 + x - ...))))))), where the unsigned coefficients in the partial numerators [3, 4, 15, 16, 35, 36,...] come in pairs of the form 4*n^2 - 1, 4*n^2 for n = 1,2,.... (End)
a(n) = Sum_{k = 1..n-1} binomial(2*n-2, 2*k-1) * a(k) * a(n-k), with a(1) = 1. - Michael Somos, Aug 02 2018
a(n) = 2^(2*n-1) * |Euler(2*n-1, 0)|, where Euler(n,x) are the Euler polynomials. - Daniel Suteu, Nov 21 2018 (restatement of one of Copeland's 2007 formulas.)
x - Sum_{n >= 1} (-1)^n*a(n)*x^(2*n)/(2*n)! = x - log(cosh(x)). The series reversion of x - log(cosh(x)) is (1/2)*x - (1/2)*log(2 - exp(x)) = Sum_{n >= 0} A000670(n)*x^(n+1)/(n+1)!. - Peter Bala, Jul 11 2022
For n > 1, a(n) = 2*Sum_{j=1..n-1} Sum_{k=1..j} binomial(2*j,j+k)*(-4*k^2)^(n-1)*(-1)^k/(4^j). - Tani Akinari, Sep 20 2023
a(n) = A110501(n) * 4^(n-1) / n (Han and Liu, 2018). - Amiram Eldar, May 17 2024

A036987 Fredholm-Rueppel sequence.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404.
Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007
a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008
a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009
a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009
A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the inverse of Sierpiński's gasket * the ruler sequence. - Gary W. Adamson, Oct 26 2009 [Of course this is only vaguely correct depending on how the fuzzy indexing in these formulas is made concrete. - R. J. Mathar, Jun 20 2014]
Characteristic function of A000225. - Reinhard Zumkeller, Mar 06 2012
Also parity of the Catalan numbers A000108. - Omar E. Pol, Jan 17 2012
For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014
Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016
Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020

			G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...
a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.

Antti Karttunen, Table of n, a(n) for n = 0..65537
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Mats Granvik, Mathematica program for computing Fredholm Rueppel sequences
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
Preda Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. doi:10.1515/crll.2004.048. MR 2076124.
H. Niederreiter and M. Vielhaber, Tree Complexity and a Doubly Exponential Gap between Structured and Random Sequences, J. Complexity, 12 (1996), 187-198.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From _Thomas Ward_, Apr 08 2009]
Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.
E. Sheppard, net.math post (1985)
Stephen Wolfram, [Page 1092] A New Kind of Science | Online.
Index entries for characteristic functions

Cf. A007404, A043545, A062518, A078885, A078585, A078886, A078887, A078888, A078889, A078890.
The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8).
Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.
If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.
This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).
Cf. A054525, A047999. - Gary W. Adamson, Oct 26 2009
Cf. A043529, A127802.

a036987 n = ibp (n+1) where
   ibp 1 = 1
   ibp n = if r > 0 then 0 else ibp n' where (n',r) = divMod n 2
a036987_list = 1 : f [0,1] where f (x:y:xs) = y : f (x:xs ++ [x,x+y])
-- Same list generator function as for a091090_list, cf. A091090.
-- Reinhard Zumkeller, May 19 2015, Apr 13 2013, Mar 13 2013

A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):
seq(A036987(n), n=0..128);

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
(* Recurrence: *)
t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] =
  If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],
   If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];
Table[t[n, k], {k, n, n}, {n, 104}]
(* Mats Granvik, Jun 03 2011 *)
mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *)
Table[PadRight[{1},2^k,0],{k,0,7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)

{a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */

a(n) = !bitand(n, n+1); \\ Ruud H.G. van Tol, Apr 05 2023

from sympy import catalan
def a(n): return catalan(n)%2 # Indranil Ghosh, May 25 2017

def A036987(n): return int(not(n&(n+1))) # Chai Wah Wu, Jul 06 2022

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
a(n) = a(floor(n/2)) * (n mod 2) for n>0 with a(0)=1. - Reinhard Zumkeller, Aug 02 2002 [Corrected by Mikhail Kurkov, Jul 16 2019]
Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...
1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003
a(n) = 1 - A043545(n). - Michael Somos, Aug 25 2003
a(n) = -Sum_{d|n+1} mu(2*d). - Benoit Cloitre, Oct 24 2003
Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
a(n) = A000108(n) mod 2 = A001405(n) mod 2. - Paul Barry, Nov 22 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006
A000523(n+1) = Sum_{k=1..n} a(k). - Mitch Harris, Jul 22 2011
a(n) = A209229(n+1). - Reinhard Zumkeller, Mar 07 2012
a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013
a(n) = A000035(A000108(n)). - Omar E. Pol, Aug 06 2013
a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014
a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015
From John M. Campbell, Jul 21 2016: (Start)
a(n) = (A000168(n-1) mod 2).
a(n) = (A000531(n+1) mod 2).
a(n) = (A000699(n+1) mod 2).
a(n) = (A000891(n) mod 2).
a(n) = (A000913(n-1) mod 2), for n>1.
a(n) = (A000917(n-1) mod 2), for n>0.
a(n) = (A001142(n) mod 2).
a(n) = (A001246(n) mod 2).
a(n) = (A001246(n) mod 4).
a(n) = (A002057(n-2) mod 2), for n>1.
a(n) = (A002430(n+1) mod 2). (End)
a(n) = 2 - A043529(n). - Antti Karttunen, Nov 19 2017
a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019
This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

Edited by M. F. Hasler, Jun 20 2014

A036279 Denominators in the Taylor series for tan(x).

Original entry on oeis.org

1, 3, 15, 315, 2835, 155925, 6081075, 638512875, 10854718875, 1856156927625, 194896477400625, 2900518163668125, 3698160658676859375, 1298054391195577640625, 263505041412702261046875, 122529844256906551386796875, 4043484860477916195764296875

Offset: 1

N. J. A. Sloane

The n-th coefficient of Taylor series for tan(x) appears to be identical to the quotient A160469(n)/A156769(n). - Johannes W. Meijer, May 24 2009

			tan(x) = x + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + ... = x + (1/3)*x^3 + (2/15)*x^5 + (17/315)*x^7 + (62/2835)*x^9 + ... =
  Sum_{n >= 1} (2^(2n) - 1) * (2x)^(2n-1) * |Bernoulli_2n| / (n*(2n-1)!).
The coefficients in the expansion of tan x are 0, 1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835, 0, 1382/155925, 0, 21844/6081075, 0, 929569/638512875, 0, ... = A002430/A036279
tanh(x) = x - (1/3)*x^3 + (2/15)*x^5 - (17/315)*x^7 + (62/2835)*x^9 - (1382/155925)*x^11 + ...
The coefficients in the expansion of tanh x are 0, 1, 0, -1/3, 0, 2/15, 0, -17/315, 0, 62/2835, 0, -1382/155925, 0, 21844/6081075, 0, -929569/638512875, 0, 6404582/10854718875, 0, -443861162/1856156927625, ... = A002430/A036279

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.67).
G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
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. 74.
H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 329.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 34, equation 34:6:1 at page 323.

Seiichi Manyama, Table of n, a(n) for n = 1..253 (first 100 terms from T. D. Noe)
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.67).
Eric Weisstein's World of Mathematics, Hyperbolic Tangent.
Eric Weisstein's World of Mathematics, Tangent.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1990. See p. 50.

Cf. A002430, A000182, A099612, A156769, A160469.

R := n -> (-1)^floor(n/2)*(4^n-2^n)*Zeta(1-n)/(n-1)!:
seq(denom(R(2*n)), n=1..18); # Peter Luschny, Aug 25 2015

f[n_] := (-1)^Floor[n/2] (4^n - 2^n) Zeta[1 - n]/(n - 1)!; Table[Denominator@ f[2 n], {n, 17}] (* Michael De Vlieger, Aug 25 2015 *)

a(n) = denominator((-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*Bernoulli(2*n)/(2*n)!). - Johannes W. Meijer, May 24 2009

Let R(x) = (cos(x*Pi/2) + sin(x*Pi/2))*(4^x-2^x)*Zeta(1-x)/(x-1)!. Then a(n) = denominator(R(2*n)) and A002430(n) = numerator(R(2*n)). - Peter Luschny, Aug 25 2015

Incorrect comment by Stephen Crowley deleted by Johannes W. Meijer, Jan 19 2009

A089171 Numerators of series coefficients of 1/(1 + cosh(sqrt(x))).

Original entry on oeis.org

1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -56963745931, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441

Offset: 0

Wouter Meeussen, Dec 07 2003

Unsigned version is equal to A002425 up to n=11, but differs beyond that point.

Unsigned version: numerators of series coefficients of 1/(1 + cos(sqrt(x))); see Mathematica. - Clark Kimberling, Dec 06 2016

N. J. A. Sloane, Table of n, a(n) for n = 0..299

Cf. A002425, A050970, A002430, A046990.
Cf. A000182, A198631, A279009, A279010.
Cf. A276592, A279370.

with(numtheory): c := n->(2^(2*n)-1)*bernoulli(2*n)/(2*n)!; seq(numer(c(n)),n=1..20); # C. Ronaldo

Numerator[CoefficientList[Series[1/(1+Cosh[Sqrt[x]]), {x, 0, 24}], x]]
Numerator[CoefficientList[Series[1/(1+Cos[Sqrt[x]]), {x, 0, 30}], x]]
(* unsigned version, Clark Kimberling, Dec 06 2016 *)

a(n) = numerator(c(n+1)) where c(n)=(2^(2*n)-1)*B(2*n)/(2*n)!, B(k) denotes the k-th Bernoulli number. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
Numerators of expansion of cosec(x)-cot(x) = 1/2*x+1/4*x^3/3!+1/2*x^5/5!+17/8*x^7/7!+31/2*x^9/9!+... - Ralf Stephan, Dec 21 2004 (Comment was applied to wrong entry, corrected by Alessandro Musesti (musesti(AT)gmail.com), Nov 02 2007)
E.g.f.: 1/sin(x)-cot(x). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/G(0); G(k) = 4*k+2-x^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: (1+x/(x-2*Q(0)))/2; Q(k) = 8*k+2+x/(1+(2*k+1)*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: x/(x+Q(0)); Q(k) = x+(x^2)/((4*k+1)*(4*k+2)-(4*k+1)*(4*k+2)/(1+(4*k+3)*(4*k+4)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: T(0)/2, where T(k) = 1 - x^2/(x^2 - (4*k+2)*(4*k+6)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013
Aerated, these are the numerators of the Taylor series coefficients of 2 * tanh(x/2) (cf. A000182 and A198631). - Tom Copeland, Oct 19 2016

A002675 Numerators of coefficients for central differences M_{4}^(2*n).

Original entry on oeis.org

1, 1, 1, 17, 31, 1, 5461, 257, 73, 1271, 60787, 241, 22369621, 617093, 49981, 16843009, 5726623061, 7957, 91625968981, 61681, 231927781, 50991843607, 499069107643, 4043309297, 1100586419201, 5664905191661, 1672180312771

Offset: 2

N. J. A. Sloane

From Peter Bala, Oct 03 2019: (Start)
Numerators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)* D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)

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).

H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
H. E. Salzer, Annotated scanned copy of left side of Table II.
H. E. Salzer, Annotated scanned copy of left side of Table III
E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource.

Cf. A002676 and A002677 (two different choices for denominators).
Also equals A002430/A002431.
Cf. A002671, A002672, A002673, A002674.

gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):
ser := series(gf, x, 40): seq(numer(coeff(ser,x,n)), n=2..28); # Peter Luschny, Oct 05 2019

More terms from Sean A. Irvine, Dec 20 2016

A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

Original entry on oeis.org

0, 1, 1, 1, 17, 31, 691, 10922, 929569, 3202291, 221930581, 9444233042, 56963745931, 29435334228302, 2093660879252671, 344502690252804724, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 28259319101491102261334882

Offset: 0

N. J. A. Sloane

			log(1/cos(x)) = 1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...
log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:3 at page 301.

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A046991, A002430, A050970.

q:= proc(n) add((-1)^k*combinat[eulerian1](n-1,k), k=0..n-1) end: A046990:= n -> numer((-1)^(n-1)*q(2*n)/(2*n)!):
seq(A046990(n),n=0..19);  # Peter Luschny, Nov 16 2012

Join[{0},Numerator[Select[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}], x],#!=0&]]] (* Harvey P. Dale, Jul 27 2011 *)
a[n_] := Numerator[((-4)^n-(-16)^n)*BernoulliB[2*n]/2/n/(2*n)!]; a[0] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 11 2014, after Charles R Greathouse IV *)

a(n)=numerator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ Charles R Greathouse IV, Nov 06 2013

{a(n) = if( n<1, 0, my(m = 2*n); numerator( polcoeff( -log(cos(x + x * O(x^m))), m)))}; /* Michael Somos, Jun 03 2019 */

# uses[eulerian1 from A173018]
def A046990(n):
    def q(n):
        return add((-1)^k*eulerian1(n-1, k) for k in (0..n-1))
    return ((-1)^(n-1)*q(2*n)/factorial(2*n)).numer()
[A046990(n) for n in (0..19)]  # Peter Luschny, Nov 16 2012

Let q(n) = Sum_{k=0..n-1} (-1)^k*A201637(n-1,k) then a(n) = numerator((-1)^(n-1)*q(2*n)/(2*n)!). - Peter Luschny, Nov 16 2012

A160469 The left hand column of the triangle A160468.

Original entry on oeis.org

1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 1936767361654, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, 1736640792209901647222, 418781231495293038913922

Offset: 1

Johannes W. Meijer, May 24 2009

Resembles A002430, the numerators of the Taylor series for tan(x). The first difference occurs at a(12). (Its resemblance to this sequence led to the conjecture A160469(n) = A002430(n)*A089170(n-1).)

Equals the first left hand column of A160468.
Equals A002430(n)*A089170(n-1).
Equals (A002430(n)/A036279(n))*(A117972(n)/A000265(n)).
Equals A048896(n-1)*A002425(n).
Cf. A156769 (which resembles the denominators of the Taylor series for tan(x)).

a(n) = A002430(n)*A089170(n-1) with A002430 (n) = numer((-1)^(n-1)*2^(2*n)*(2^(2*n)-1)* bernoulli(2*n)/(2*n)!) and A089170 (n-1) = numer(2*bernoulli(2*n)* (4^n-1)/(2*n))/ numer((4^n-1)*bernoulli(2*n)/(2*n)!) for n = 1, 2, 3, ....

cp's OEIS Frontend

A092505 a(n) = A002430(n) / A046990(n).

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A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

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A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

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A000182 Tangent (or "Zag") numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).

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A036987 Fredholm-Rueppel sequence.

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A036279 Denominators in the Taylor series for tan(x).

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