A001586 -id:A001586 - cp's OEIS frontend

A201923 E.g.f. satisfies: A(x) = 1/(cos(xA(x)) - sin(xA(x))).

1, 1, 5, 44, 581, 10256, 227529, 6088256, 190930729, 6870227200, 279066777613, 12632667642880, 630670054092525, 34426087332253696, 2039903110075608977, 130404672744539242496, 8946117466489960168913, 655585000075494566199296, 51111210765059412626238741

Offset: 0

Paul D. Hanna, Dec 06 2011

nonn

Compare e.g.f. to: LambertW(-x)/(-x) = (1/x)*Series_Reversion(x*(cosh(x) - sinh(x))).
The radius of convergence r of e.g.f. A(x) is given by:
r = t*(cos(t) - sin(t)) where tan(t) = (1-t)/(1+t), which evaluates as:
r = 0.21266685344074710045360679397024815598865409988038...
t = 0.40262817418811160981993252391123072456350647779608...
Further, A(r) = 1/(cos(t) - sin(t)), thus
A(r) = 1.89323426605496483543109751303457163422769666683274...

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 44*x^3/3! + 581*x^4/4! + 10256*x^5/5! +...
where
1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! +...+ A001586(n)*x^n/n! +...
The coefficient of x^n/n! in powers of G(x) = 1/(cos(x)-sin(x)) begins:
G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
G^2: [1,(2), 8, 40, 256, 1952, 17408, 177280, ..., A000828(n+1), ...];
G^3: [1, 3,(15), 93, 705, 6243, 63375, 724413, ...];
G^4: [1, 4, 24,(176), 1536, 15424, 175104, 2214656, ...];
G^5: [1, 5, 35, 295,(2905), 32525, 407435, 5638495, ...];
G^6: [1, 6, 48, 456, 4992, (61536), 841728, 12633216, ...];
G^7: [1, 7, 63, 665, 8001, 107527, (1592703), 25738265, ...];
G^8: [1, 8, 80, 928, 12160, 176768, 2816000, (48706048), ...]; ...
where coefficients in parenthesis form the initial terms of this sequence:
[1/1, 2/2, 15/3, 176/4, 2905/5, 61536/6, 1592703/7, 48706048/8, ...].

Cf. A001586.

CoefficientList[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(1/x*serreverse(x*(cos(X)-sin(X) )),n)}

{a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(cos(X*A) - sin(X*A)));n!*polcoeff(A,n)}

E.g.f. satisfies: A( x*(cos(x) - sin(x)) ) = 1/(cos(x) - sin(x)).
E.g.f: (1/x) * Series_Reversion( x*(cos(x) - sin(x)) ).
a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(n+1) / (n+1).
a(n) ~ n^(n-1) * sqrt((t*cos(2*t))/(3+sin(2*t))) / (exp(n) * r^(n+1)), where r and t were described above. - Vaclav Kotesovec, Jan 12 2014

A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065

Offset: 0

Seiichi Manyama, Jun 24 2025

sign

Cf. A352252, A385284.
Cf. A001586, A235134, A380155, A385281.
Cf. A001147, A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*(2*I)^(n-k)*a185951(n, k));

a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).

Original entry on oeis.org

1, 0, 2, -4, 28, -160, 1272, -11184, 114448, -1309440, 16680992, -233587264, 3569157568, -59075960320, 1053056675712, -20111857791744, 409715696197888, -8868323731660800, 203247024658514432, -4916860703228314624, 125206830774036241408, -3347784042587048058880

Offset: 0

Paul Barry, Mar 16 2006

Row sums of number triangle A117442. Binomial transform of alternating sign Springer numbers (-1)^n*A001586(n).

G. C. Greubel, Table of n, a(n) for n = 0..425

Cf. A117442.

CoefficientList[Series[E^x/(Cos[x]+Sin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
A117442[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Abs[Numerator[EulerE[n-k, 1/4]]]; Table[Sum[A117442[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jun 02 2021 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(x)/(cos(x) + sin(x)))) \\ Michel Marcus, Jun 02 2021

@CachedFunction
def f(n): return (-1/4)^n*sum( binomial(n, j)*2^j*euler_number(j) for j in (0..n) ) # f(n) = Euler(n, 1/4)
def A117443(n): return sum( (-1)^(n+k)*binomial(n,k)*abs(numerator(f(n-k))) for k in (0..n) )
[A117443(n) for n in (0..30)] # G. C. Greubel, Jun 02 2021

E.g.f.: 1/Q(0); Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 28 2011
G.f.: 1/Q(0) where Q(k) = 1 + 4*k*x - 2*x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - 2*x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 10 2013
a(n) ~ (-1)^n * n! * 2^(2*n+3/2) / (Pi^(n+1) * exp(Pi/4)). - Vaclav Kotesovec, Aug 04 2014
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * abs(numerator( Euler(n-k, 1/4) )), where Euler(n, x) is the Euler number polynomial. - G. C. Greubel, Jun 02 2021

A235166 E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.

Original entry on oeis.org

1, 1, 4, 16, 88, 640, 5440, 54400, 620800, 7966720, 113651200, 1783091200, 30519808000, 565916876800, 11300689100800, 241781039104000, 5517822373888000, 133795711025152000, 3435107208822784000, 93093522064998400000, 2655675672405606400000, 79546285618254315520000

Offset: 0

Paul D. Hanna, Jan 04 2014

nonn

See comments by Roland Bacher in A098777 which imply that this sequence is related to elliptic functions.

Compare to: G'(x) = G(x)^2/G(-x) dx, which holds when G(x) = 1/(cos(x) - sin(x)), the e.g.f. of A001586 (Springer numbers).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 88*x^4/4! + 640*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 10*x^2/2! + 56*x^3/3! + 400*x^4/4! + 3440*x^5/5! +...
1/A(x) = 1 - x - 2*x^2/2! + 2*x^3/3! + 16*x^4/4! - 40*x^5/5! - 320*x^6/6! +...+ A098777(n)*x^n/n! +...

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Cf. A001586, A098777.

kmax = 21;
A[x_] = 1+x; Do[A[x_] = 1+Integrate[A[x]^2/A[-x]^2+O[x]^k, x] // Normal, {k, 1, kmax}];
CoefficientList[A[x], x] Range[0, kmax]! (* Jean-François Alcover, Jul 29 2018 *)

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(A^2/subst(A, x,-x +x*O(x^n))^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

{a(n)=local(A=1); A=1/(1-3*serreverse(intformal(1/(1-9*x^2 +x*O(x^n))^(2/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/(1 - 3*Series_Reversion( Integral 1/(1 - 9*x^2)^(2/3) dx ))^(1/3).
E.g.f.: 1/F(x), where F(x) equals the e.g.f. of A098777 (pseudo-factorials).
a(n) ~ 2^(-2/3) * n! * (9*GAMMA(2/3)^3/(2^(2/3)*Pi^2))^(n+1). - Vaclav Kotesovec, Feb 24 2014

A245204 The unique integer r with |r| < prime(n)/2 such that E_{prime(n)-3}(1/4) == r (mod prime(n)), where E_m(x) denotes the Euler polynomial of degree m.

Original entry on oeis.org

1, 2, 2, 4, 1, 1, 5, 1, -2, -6, 10, 14, 5, 7, 7, -28, -12, 13, 14, 26, -21, -31, -13, -10, -11, -7, -6, 5, 2, -21, 2, 33, -15, -24, 34, 71, -15, 24, 9, 37, 73, -18, -84, -65, 9, -90, -65, -47, 97, -64, -100, -8, 41, 81, -81, -71, -65, -70, 113, 10, -80, 119, 57, 78, 20, 124, 167, -71, -48

Offset: 2

Zhi-Wei Sun, Jul 13 2014

sign

Conjecture: a(n) = 0 infinitely often. In other words, there are infinitely many odd primes p such that E_{p-3}(1/4) == 0 (mod p) (equivalently, p divides A001586(p-3)).
This seems reasonable in view of the standard heuristic arguments. The first n with a(n) = 0 is 171 with prime(171) = 1019. The next such a number n is greater than 2600 and hence prime(n) > 23321.
Zhi-Wei Sun made many conjectures on congruences involving E_{p-3}(1/4), see the reference.

			a(3) = 2 since E_{prime(3)-3}(1/4) = E_2(1/4) = -3/16 == 2 (mod prime(3)=5).

Zhi-Wei Sun, Table of n, a(n) for n = 2..1300
Zhi-Wei Sun, Super congruences and Euler numbers, arXiv:1001.4453 [math.NT].
Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math. 54(2011), 2509-2535.

Cf. A000040, A001586, A122045, A198245, A245089, A245206.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
a[n_]:=rMod[EulerE[Prime[n]-3,1/4],Prime[n]]
Table[a[n],{n,2,70}]

A339058 a(n) = 4^nEuler(n, 1/4)2^(valuation_{2}(n + 1)).

Original entry on oeis.org

1, -2, -3, 44, 57, -722, -2763, 196888, 250737, -5746082, -36581523, 2049374444, 7828053417, -259141449842, -2309644635483, 705775346640176, 898621108880097, -38901437271432002, -445777636063460643, 43136210244502819244, 274613643571568682777, -14685255919931552812562

Offset: 0

Peter Luschny, Nov 27 2020

sign

			The array of the general case starts:
[k]
[1] 1,  1,  0, -1,   0,     1,     0,     -17,       0, ... [A198631]
[2] 1,  0, -1,  0,   5,     0,   -61,       0,    1385, ... [A122045]
[3] 1, -1, -2, 13,  22,  -121,  -602,   18581,   30742, ... [A156179]
[4] 1, -2, -3, 44,  57,  -722, -2763,  196888,  250737, ... [this sequence]
[5] 1, -3, -4, 99, 116, -2523, -8764, 1074243, 1242356, ... [A156182]
...

Michael De Vlieger, Table of n, a(n) for n = 0..432

Cf. A198631, A122045, A156179, A156182, A156191, A339057.

Note the difference from A001586, A188458, and A212435.

a := n -> 4^n*euler(n, 1/4)*2^padic[ordp](n+1, 2): seq(a(n), n=0..9);

Array[4^#*EulerE[#, 1/4]*2^IntegerExponent[# + 1, 2] &, 22, 0] (* Michael De Vlieger, Mar 15 2022 *)

def euler_sum(n):
    return (-1)^n*sum(2^k*binomial(n, k)*euler_number(k) for k in (0..n))
def a(n): return euler_sum(n) << valuation(n + 1, 2)
print([a(n) for n in range(22)])

A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).

Original entry on oeis.org

1, 1, 3, 14, 81, 616, 5523, 58064, 697281, 9417856, 141368643, 2334020864, 42039523281, 820296426496, 17237259945363, 388087200241664, 9320064293358081, 237814050877505536, 6425096888209255683, 183232685725482942464, 5500505587921088841681

Offset: 0

Seiichi Manyama, Mar 06 2022

nonn

Cf. A000182, A000828, A001586, A175288, A352164.

m = 20; Range[0, m]! * CoefficientList[Series[1/(Cos[x] - Tan[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(cos(x)-tan(x))))

c(n) = ((-4)^n-(-16)^n)*bernfrac(2*n)/(2*n);
b(n) = if(n%2==1, c((n+1)/2), (-1)^(n/2+1));
a(n) = if(n==0, 1, sum(k=1, n, b(k)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = Sum_{k=1..n} b(k) * binomial(n,k) * a(n-k), where b(k) = A000182((k+1)/2) if k is odd, otherwise (-1)^(k/2+1).
From Vaclav Kotesovec, Mar 06 2022: (Start)
a(n) ~ n! / (sqrt(5) * (arctan(sqrt((sqrt(5) - 1)/2)))^(n+1)).
a(n) ~ n! / (sqrt(5) * A175288^(n+1)). (End)

A385896 Array read by ascending antidiagonals: A(n, k) = k! * [x^k] (1 - sin(n*x))^(-1/n) for n > 0, A(0, k) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 11, 16, 1, 1, 1, 5, 19, 57, 61, 1, 1, 1, 6, 29, 136, 361, 272, 1, 1, 1, 7, 41, 265, 1201, 2763, 1385, 1, 1, 1, 8, 55, 456, 3001, 13024, 24611, 7936, 1, 1, 1, 9, 71, 721, 6301, 42125, 165619, 250737, 50521, 1

Offset: 0

Peter Luschny, Jul 20 2025

			Table starts:
  [0] 1, 1, 1,  1,    1,     1,      1, ... [A000012]
  [1] 1, 1, 2,  5,   16,    61,    272, ... [A000111]
  [2] 1, 1, 3, 11,   57,   361,   2763, ... [A001586]
  [3] 1, 1, 4, 19,  136,  1201,  13024, ... [A007788]
  [4] 1, 1, 5, 29,  265,  3001,  42125, ... [A144015]
  [5] 1, 1, 6, 41,  456,  6301, 108576, ... [A230134]
  [6] 1, 1, 7, 55,  721, 11761, 240247, ... [A227544]
  [7] 1, 1, 8, 71, 1072, 20161, 476288, ... [A235128]
  [8] 1, 1, 9, 89, 1521, 32401, 869049, ... [A230114]
     [A000027]  | [A187277] | [A385898].
            [A028387]   [A385897]
.
Seen as a triangle:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 2,  1;
  [4] 1, 1, 3,  5,   1;
  [5] 1, 1, 4, 11,  16,    1;
  [6] 1, 1, 5, 19,  57,   61,    1;
  [7] 1, 1, 6, 29, 136,  361,  272,    1;
  [8] 1, 1, 7, 41, 265, 1201, 2763, 1385, 1;

Rows: A000012, A000111, A001586, A007788, A144015, A230134, A227544, A235128, A230114.
Columns: A000027, A028387, A187277, A385897, A385898.
Cf. A135278.

MAX := 16: ser := n -> series((1 - sin(n*x))^(-1/n), x, MAX):
A := (n, k) -> if n = 0 then 1 else k!*coeff(ser(n), x, k) fi:
seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);

T[n_, k_, m_] := T[n, k, m] =
  Which[
    n <  0 || k <  0, 0,
    n == 0 && k == 0, 1,
    k == 0, T[n - 1, n - 1, m],
    True, T[n, k - 1, m] + m*T[n - 1, n - k - 1, m]
];
A[n_, k_] := T[k, k, n - k];
Table[A[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

A(n, k) = T(k, k, n - k) where T(n, k, m) = T(n, k-1, m) + m * T(n-1, n-k-1, m) for k > 0, T(n, 0, m) = T(n-1, n-1, m), and T(0, 0, m) = 1.

Column n is a linear recurrence with kernel [(-1)^k*A135278(n, k), k = 0..n].

A245206 Odd primes p with E_{p-3}(1/4) == 0 (mod p), where E_n(x) denotes the Euler polynomial of degree n.

Original entry on oeis.org

1019

Offset: 1

Zhi-Wei Sun, Jul 13 2014

The conjecture in A245204 asserts that the current sequence contains infinitely many primes.

Our computation shows that the second term should be greater than prime(2600) = 23321.

			a(1) = 1019 since 1019 is a prime with E_{1019-3}(1/4) == 88*1019 (mod 1019^2).

Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math., Vol. 54, No. 12 (2011), 2509-2535; arXiv preprint, arXiv:1001.4453 [math.NT], 2010-2011.
Index entries for one-term sequences.

Cf. A000040, A001586, A122045, A198245, A245089, A245204.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
n=0;Do[If[rMod[EulerE[Prime[k]-3,1/4],Prime[k]]==0,n=n+1;Print[n," ",Prime[k]]],{k,2,200}]

A343171 Irregular triangle read by rows: coefficients of polynomials xi_n.

Original entry on oeis.org

1, 1, 3, 1, 11, 5, 57, 38, 5, 361, 302, 61, 2763, 2827, 845, 61, 24611, 29607, 11421, 1385, 250737, 347372, 165678, 30108, 1385, 2873041, 4501564, 2551326, 610444, 50521, 36581523, 63967093, 42044902, 12558738, 1578727, 50521

Offset: 0

N. J. A. Sloane, Apr 21 2021

			Triangle begins:
    1;
    1;
    3,   1;
   11,   5;
   57,  38,  5;
  361, 302, 61;
  ...

Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374 [math.CO], 2021.

First column gives A001586.

Cf. A343170.

xi[0][_] = 1;
xi[n_][x_] := xi[n, x] = (2n - 1 + (n-1) x) xi[n-1][x] - (1 + x)(4 + 2x)* xi[n-1]'[x];
Table[CoefficientList[xi[n][x], x], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 21 2021 *)

More terms from Jean-François Alcover, Apr 21 2021

cp's OEIS Frontend

A201923 E.g.f. satisfies: A(x) = 1/(cos(x*A(x)) - sin(x*A(x))).

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A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

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A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).

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A235166 E.g.f. satisfies: A'(x) = A(x)^2/A(-x)^2, with A(0)=1.

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A245204 The unique integer r with |r| < prime(n)/2 such that E_{prime(n)-3}(1/4) == r (mod prime(n)), where E_m(x) denotes the Euler polynomial of degree m.

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A339058 a(n) = 4^n*Euler(n, 1/4)*2^(valuation_{2}(n + 1)).

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A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).

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A385896 Array read by ascending antidiagonals: A(n, k) = k! * [x^k] (1 - sin(n*x))^(-1/n) for n > 0, A(0, k) = 1.

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A201923 E.g.f. satisfies: A(x) = 1/(cos(xA(x)) - sin(xA(x))).

A339058 a(n) = 4^nEuler(n, 1/4)2^(valuation_{2}(n + 1)).