A001586 -id:A001586 - cp's OEIS frontend

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

1, 1, 3, 19, 153, 1561, 19563, 289339, 4932273, 95258161, 2055639123, 49019157859, 1280056939593, 36329281202761, 1113449691889083, 36651273215389579, 1289577677407798113, 48299079453732363361, 1918528841276621473443, 80559757274836073592499

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

G. C. Greubel, Table of n, a(n) for n = 0..395

Cf. A006154, A235135.

Cf. A001147, A001586, A136630, A235131, A380155.

CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ G. C. Greubel, Apr 05 2017

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A117442 Number triangle read by rows, related to exp(x)/(cos(x) + sin(x)).

Original entry on oeis.org

1, -1, 1, 3, -2, 1, -11, 9, -3, 1, 57, -44, 18, -4, 1, -361, 285, -110, 30, -5, 1, 2763, -2166, 855, -220, 45, -6, 1, -24611, 19341, -7581, 1995, -385, 63, -7, 1, 250737, -196888, 77364, -20216, 3990, -616, 84, -8, 1, -2873041, 2256633, -885996, 232092, -45486, 7182, -924, 108, -9, 1

Offset: 0

Paul Barry, Mar 16 2006

			Triangle begins
       1;
      -1,     1;
       3,    -2,     1;
     -11,     9,    -3,    1;
      57,   -44,    18,   -4,    1;
    -361,   285,  -110,   30,   -5,  1;
    2763, -2166,   855, -220,   45, -6,  1;
  -24611, 19341, -7581, 1995, -385, 63, -7, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Inverse of A117440.
Second column contains A161722 as subsequence.
Cf. A000027, A001586, A045943, A111080, A122045, A117443 (row sums).

A117442_row := proc(n) 2^n*add(binomial(n,k)*euler(k)*((x+1)/2)^(n-k), k=0..n);
seq((-1)^(n-j)*abs(coeff(%,x,j)),j=0..n) end:
seq(print(A117442_row(n)),n=0..5);  # Peter Luschny, Jun 08 2013

row[n_] := row[n] = 2^n Sum[Binomial[n, k] EulerE[k] ((x+1)/2)^(n-k), {k, 0, n}];
T[n_, k_] := (-1)^(n-k) Abs[Coefficient[row[n], x, k]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019, from Maple *)
Table[(-1)^(n-k)*Binomial[n, k]*Abs[Numerator[EulerE[n-k, 1/4]]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 02 2021 *)

E(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1); \\ A122045
p(n) = 2^n*sum(k=0, n, binomial(n,k)*E(k)*((x+1)/2)^(n-k));
row(n) = my(rp=p(n)); vector(n+1, k, k--; (-1)^(n-k)*abs(polcoeff(rp, k))); \\ Michel Marcus, Nov 16 2020

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 A117442(n,k): return (-1)^(n+k)*binomial(n,k)*abs(numerator(f(n-k)))
flatten([[A117442(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 02 2021

T(n, 0) = (-1)^n*A001586(n).
Sum_{k=0..n} T(n, k) = A117443(n).
Column k has e.g.f. (x^k/k!)/(cos(x) + sin(x)).
Apart from signs the T(n,k) are the coefficients of the polynomials p(n, x) = 2^n*Sum_{k=0..n} binomial(n,k)*euler(k)*((x+1)/2)^(n-k). - Peter Luschny, Jun 08 2013
From G. C. Greubel, Jun 02 2021: (Start)
T(n, k) = (-1)^(n+k) * binomial(n, k) * abs(numerator( Euler(n-k, 1/4) )), where Euler(n, x) is the Euler number polynomial.
T(n, n) = 1.
T(n, n-1) = -A000027(n) = -binomial(n+1, 1).
T(n, n-2) = A045943(n+1) = 3*binomial(n+2, 2).
T(n, n-3) = -A111080(n) = -11*binomial(n+3, 3).
T(j, k) = (-1)^k * binomial(j+k, k) * abs(numerator( Euler(k, 1/4) )) (columns).
T(n, n-j) = (-1)^n * binomial(n+j, j) * abs(numerator( Euler(n, 1/4) )) (downward diagonals). (End)
The pair of triangles P*((I + P^4)/2)^(-1) and P^3*((I + P^4)/2)^(-1), where P denotes Pascal's triangle A007318, give the present triangle but with a different pattern of signs. - Peter Bala, Mar 07 2024

A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

Original entry on oeis.org

1, 1, 8, 71, 1072, 20161, 476288, 13315751, 432387712, 15959926081, 660372282368, 30265936565831, 1522069164439552, 83327826089289601, 4933286107483701248, 314052936209639958311, 21392225375507849838592, 1552501782546292090638721, 119588747474281844162428928

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)).

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A230114 (p=8).

Cf. A045754, A136630.

CoefficientList[Series[1/(1-Sin[7*x])^(1/7), {x, 0, 20}], x] * Range[0, 20]!

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045754(n) = prod(k=0, n-1, 7*k+1);
a(n) = sum(k=0, n, a045754(k)*(7*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 2^(n+3/7) * 7^n / (Gamma(2/7) * n^(5/7) * Pi^(n+2/7)).

a(n) = Sum_{k=0..n} A045754(k) * (7*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A318005 E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).

Original entry on oeis.org

1, 4, 24, 224, 2880, 48064, 989184, 24218624, 687083520, 22151148544, 799546834944, 31934834253824, 1398132497448960, 66573473015578624, 3425078687463112704, 189331392774496845824, 11190654534195295027200, 704262689221037166690304, 47015904809670036594622464, 3318579148264602406039322624

Offset: 1

Paul D. Hanna, Aug 27 2018

nonn

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 48064*x^6/6! + 989184*x^7/7! + 24218624*x^8/8! + 687083520*x^9/9! + 22151148544*x^10/10! + ...
such that:
cos(A(x)) + sin(A(x)) = 1/( cos(x) - sin(x) ).
RELATED SERIES.
(a) cos(A(x)) + sin(A(x)) = 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! + ...
(b) If F(F(x)) = A(x), then
F(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 360*x^5/5! + 4592*x^6/6! + 70896*x^7/7! + 1279360*x^8/8! + ... + A318006(n)*x^n/n! + ...
where F(x) = arcsin( 2*sin(2*x)/(2 - sin(2*x)) ) /2.

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A318006, A318000, A200560.

{a(n) = my(A = asin( sin(2*x +x*O(x^n))/(1 - sin(2*x +x*O(x^n))) )/2 ); n!*polcoeff(A,n)}
for(n=1,20, print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(-A(-x)) = x.
(2) 1 = Sum_{n>=0} (-1)^floor(n/2) * ( A(x) + (-1)^n*x )^n/n!.
(3a) 1 = cos(A(x) + x) + sin(A(x) - x).
(3b) 1 = ( cos(A(x)) + sin(A(x)) ) * ( cos(x) - sin(x) ).
(4) A(x) = arcsin( sin(2*x)/(1 - sin(2*x)) )/2.
a(n) = 2^(n-1) * A200560(n).

A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 27, 249, 2825, 41355, 708883, 13888497, 309267729, 7698772755, 211585744139, 6367841422569, 208299923870233, 7357493992966299, 279095125351544835, 11316313498670411745, 488403056864943302177, 22355228989851909617187, 1081663315375339026249211

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A205571, A385282.
Cf. A001586, A235134, A380155, A385283.
Cf. A001147, A185951.
Cf. A069814.

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^(n-k)*a185951(n, k));

a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.

a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + r*sqrt(1 - r^2)) * exp(n) * r^n), where r = A069814. - Vaclav Kotesovec, Jun 24 2025

A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 14, 93, 796, 8343, 103424, 1479993, 24008656, 435364683, 8726775584, 191601310293, 4572794295616, 117871476051423, 3263515787807744, 96591500816346993, 3043368045293138176, 101702692426476460563, 3592948632452749243904, 133794496537591022166093

Offset: 0

Seiichi Manyama, Jun 24 2025

nonn

Cf. A000111, A385307.
Cf. A380015, A385304, A385308, A385310.
Cf. A001147, A001586, A007289, A136630.

With[{nn=20},CoefficientList[Series[1/Sqrt[1-2Sin[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2025 *)

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*I^(n-k)*a136630(n, k));

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

a(n) ~ 2^(n+1) * 3^(n + 1/4) * n^n / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025

A079858 E.g.f. 1/(cos(2x) - sin(2x)).

Original entry on oeis.org

1, 2, 12, 88, 912, 11552, 176832, 3150208, 64188672, 1470996992, 37459479552, 1049279715328, 32063706796032, 1061443378552832, 37841217707753472, 1445427909919080448, 58892032991566036992, 2549444593020567683072

Offset: 0

Michael Somos, Jan 20 2003

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A001586.

(-1)^(n*(n-1)/2)*a(n) gives the alternating row sums of A225118. - Wolfdieter Lang, Jul 12 2017

CoefficientList[Series[1/(Cos[2*x]-Sin[2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

a(n)=if(n<0,0,n!*polcoeff(1/(cos(2*x+x*O(x^n))-sin(2*x+x*O(x^n))),n))

x='x+O('x^66); v=Vec(serlaplace( 1/(cos(2*x)-sin(2*x)) ) ) \\ Joerg Arndt, Apr 27 2013

from mpmath import mp, lerchphi
mp.dps = 32; mp.pretty = True
def A079858(n): return abs(2*8^n*lerchphi(-1,-n,1/4))
[int(A079858(n)) for n in (0..17)]  # Peter Luschny, Apr 27 2013

E.g.f.: 1/(cos(2*x) - sin(2*x)).
a(n) = 2^n * A001586(n).
a(n) = | 2*8^n*lerchphi(-1,-n,1/4) |. - Peter Luschny, Apr 27 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 8*x^2*(k+1)^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013
a(n) ~ 4 * n^(n+1/2) * (8/Pi)^n / (sqrt(Pi)*exp(n)). - Vaclav Kotesovec, Oct 07 2013
E.g.f.:  1/(1-2*x)*(1 + 2*x^2/((1-2*x)*W(0) - x )), where W(k) = x + (k+1)/( 1 - 2*x/( 2*k+3 - x*(2*k+3)/W(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Dec 27 2013

A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

Original entry on oeis.org

1, 8, 7, 128, 304, 177, 3072, 13952, 21080, 10199, 98304, 724992, 2016000, 2441056, 1051745, 3932160, 42762240, 187643904, 407505664, 428605352, 169913511, 188743680, 2839019520, 17974591488, 60428242944, 111985428352

Offset: 0

Ralf Stephan, Sep 07 2004

			S(0,x) = 1,
S(1,x) = 8*x + 7,
S(2,x) = 128*x^2 + 304*x + 177,
S(3,x) = 3072*x^3 + 13952*x^2 + 21080*x + 10199.

A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A001586. S(n, 1/2) = A000464(n+1), S(n, -1/2) = A000281(n).
Leading coefficients are A051189. Constant terms are in A098433.
Cf. A001586. S(n, 1/2) = A000464(n), S(n, -1/2) = A000281(n).

S(n,x)=if(n<1,1,(2*x+2)*(2*x+4)*S(n-1,x+2)-(2*x+1)^2*S(n-1,x))

Recurrence: S(0, x)=1, S(n, x)=(2x+2)(2x+4)S(n-1, x+2)-(2x+1)^2S(n-1, x).

G.f.: Sum[n>=0, S(n, x)t^n] = 1/(1+t-4*2(x+1)t/(1-4*2(x+2)t/(1+t-4*4(x+3)t/(1-4+4(x+4)t/...)))).

A151751 Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.

Original entry on oeis.org

2, 0, 6, -44, 0, 12, 0, -220, 0, 20, 2166, 0, -660, 0, 30, 0, 15162, 0, -1540, 0, 42, -196888, 0, 60648, 0, -3080, 0, 56, 0, -1771992, 0, 181944, 0, -5544, 0, 72, 28730410, 0, -8859960, 0, 454860, 0, -9240, 0, 90

Offset: 2

Peter Bala, Jun 17 2009

Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * sum {r = 0..m-1} X(r)*exp(r*t)
= sum {n = 0..inf} B_n(X,x)*t^n/n!.
For the theory and properties of these polynomials see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
Cf. A153641.

			The triangle begins
n\k|........0.......1........2.......3......4.......5.......6
=============================================================
.2.|........2
.3.|........0.......6
.4.|......-44.......0.......12
.5.|........0....-220........0......20
.6.|.....2166.......0.....-660.......0......30
.7.|........0...15162........0...-1540.......0.....42
.8.|..-196888.......0....60648.......0...-3080......0......56
...

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.

with(gfun):
for n from 2 to 10 do
Genbernoulli(n,x) := 8^(n-1)*(bernoulli(n,(x+1)/8)-bernoulli(n,(x+3)/8)-bernoulli(n,(x+5)/8)+bernoulli(n,(x+7)/8));
seriestolist(series(Genbernoulli(n,x),x,10))
end do;

TABLE ENTRIES
(1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;
(2)... T(2*n,2*k) = (-1)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);
(3)... T(2*n+1,2*k+1) = (-1)^(n-k-1)*C(2*n+1,2*k+1)*2*(n-k)*A000464(n-k-1);
where C(n,k) = binomial(n,k).
GENERATING FUNCTION
The e.g.f. for these generalized Bernoulli polynomials is
(4)... t*exp(x*t)*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1)
= sum {n = 2..inf} B_n(X,x)*t^n/n! = 2*t^2/2! + 6*x*t^3/3! + (12*x^2 - 44)*t^4/4! + ....
In terms of the ordinary Bernoulli polynomials B_n(x)
(5)... B_n(X,x) = 8^(n-1)*{B_n((x+1)/8) - B_n((x+3)/8) - B_n((x+5)/8) + B_n((x+7)/8)}.
The B_n(X,x) are Appell polynomials of the form
(6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(n-j).
The sequence of generalized Bernoulli numbers
(7)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(8)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(9)... t*sinh(t)/cosh(2*t).
Hence
(10)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1); B_(2*n+1)(X,0) = 0.
The sequence {B_(2*n)(X,0)}n>=2 is A161722.
RELATION WITH TWISTED SUMS OF POWERS
The generalized Bernoulli polynomials may be used to evaluate sums of k-th powers twisted by the function X(n). For the present case the result is
(11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^k
= [B_(k+1)(X,8*N) - B_(k+1)(X,0)]/(k+1)
For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].
MISCELLANEOUS
(12)... Row sums [2, 6, -32, ...] = (-1)^(1+binomial(n,2))*A109572(n)
= (-1)^(1+binomial(n,2))*n*A000828(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).

A190392 E.g.f. A(x) satisfies A'(x) = sin(A(x)) + cos(A(x)).

Original entry on oeis.org

1, 1, 0, -4, -12, 4, 240, 1184, -1008, -59504, -401280, 643136, 38584128, 323581504, -848090880, -51666451456, -509739310848, 2004840714496, 123888658698240, 1386061762251776, -7721141999864832, -483475390212586496, -5974101514137292800, 45231727252157947904

Offset: 1

Vladimir Kruchinin, May 09 2011

sign

Let f(x) be a smooth function. The autonomous differential equation A'(x) = f(A(x)), with initial condition A(0) = 0, is separable and the solution is given by A(x) = inverse function of Integral_{t = 0..x} 1/f(t) dt. The inversion of the integral Integral_{t = 0..x} 1/f(t) dt is most conveniently found by applying [Dominici, Theorem 4.1]. The result is A(x) = Sum_{n>=1} D^(n-1)[f](0)*x^n/n!, where the nested derivative D^n[f](x)is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = (d/dx)(f(x)*D^n[f](x)) for n >= 0. See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x). In the present case we take f(x) = sin(x)+cos(x). - Peter Bala, Aug 27 2011

Alois P. Heinz, Table of n, a(n) for n = 1..200
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.

Cf. A001586, A012244, A028296.

A := x ; for i from 1 to 35 do sin(A)+cos(A) ; convert(taylor(%,x=0,25),polynom) ; A := int(%,x) ; print(A) ; end do:
for i from 1 to 25 do printf("%d,", coeftayl(A,x=0,i)*i!) ; end do: # R. J. Mathar, Jun 03 2011

terms = 25; A[] = 0; Do[A[x] = Integrate[Sin[A[x]] + Cos[A[x]], x] + O[x]^terms, terms]; CoefficientList[A[x], x]*Range[0, terms-1]! (* Jean-François Alcover, Feb 21 2013, updated Jan 15 2018 *)

g(n):=(-1)^floor(n/2)*1/n!;
a(n):=T190015_Solve(n,g);

E.g.f.: A(x) = inverse of Integral_{t = 0..x} 1/(sin(t)+cos(t)) dt = series reversion (x - x^2/2! + 3*x^3/3! - 11*x^4/4! + 57*x^5/5! - ...) = x + x^2/2! - 4*x^4/4! - 12*x^5/5! + .... a(n) = D^(n-1)[sin(x)+cos(x)](0), where the nested derivative operator D^n is defined above. Compare with A012244. -Peter Bala, Aug 27 2011
E.g.f.: A(x) = 2*arctan((sqrt(2)-1)*exp(sqrt(2)*x))-Pi/4. Compare with A028296. - Peter Bala, Sep 02 2011
G.f.: 1/G(0) where G(k) = 1 - 2*x*(k+1)/(1 + 1/(1 - 2*x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 10 2013.
G.f.: -(1/x)/Q(0), where Q(k)= 2*k+1 - 1/x + (k+1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 + (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
E.g.f. if offset 0: 2^(1/2)/(2^(1/2)*cosh(x*2^(1/2))-sinh(x*2^(1/2))). - Sergei N. Gladkovskii, Nov 10 2013
a(n) ~ -(n-1)! * 2^(3*n/2+1) * sin(n*arctan(Pi/log(3 - 2*sqrt(2)))) / (Pi^2 + log(3 - 2*sqrt(2))^2)^(n/2). - Vaclav Kotesovec, Jan 07 2014

cp's OEIS Frontend

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

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A117442 Number triangle read by rows, related to exp(x)/(cos(x) + sin(x)).

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A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

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A318005 E.g.f.: A(x) satisfies: cos(A(x)) + sin(A(x)) = 1/(cos(x) - sin(x)).

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A385281 Expansion of e.g.f. 1/(1 - 2 * x * cosh(2*x))^(1/2).

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A385306 Expansion of e.g.f. 1/(1 - 2 * sin(x))^(1/2).

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A079858 E.g.f. 1/(cos(2*x) - sin(2*x)).

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A098432 Coefficients of polynomials S(n,x) related to Springer numbers.

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A079858 E.g.f. 1/(cos(2x) - sin(2x)).