A153641 -id:A153641 - cp's OEIS frontend

A225147 a(n) = Im((1-I)^(1-n)*A_{n, 3}(I)) where A_{n, k}(x) are the generalized Eulerian polynomials.

-1, 2, 5, -46, -205, 3362, 22265, -515086, -4544185, 135274562, 1491632525, -54276473326, -718181418565, 30884386347362, 476768795646785, -23657073914466766, -417370516232719345, 23471059057478981762, 465849831125196593045, -29279357851856595135406

Offset: 0

Peter Luschny, Apr 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Generalized Eulerian polynomials.

Cf. A000810 (real part (up to sign)), A212435 (k=2), A122045 (k=1), A002439.

Bisections are A002438 and A000191.

B := proc(n, u, k) option remember;
if n = 1 then if (u < 0) or (u >= 1) then 0 else 1 fi
else k*u*B(n-1, u, k) + k*(n-u)*B(n-1, u-1, k) fi end:
EulerianPolynomial := proc(n, k, x) local m; if x = 0 then RETURN(1) fi;
add(B(n+1, m+1/k, k)*u^m, m = 0..n); subs(u=x, %) end:
seq(Im((1-I)^(1-n)*EulerianPolynomial(n, 3, I)), n=0..19);

CoefficientList[Series[-E^(-2*x)*Sech[3*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 29 2014 after Sergei N. Gladkovskii *)
Table[-6^n EulerE[n,1/6], {n,0,19}] (* Peter Luschny, Nov 16 2016 after Peter Bala *)

from mpmath import mp, polylog, im
mp.dps = 32; mp.pretty = True
def A225147(n): return im(-2*I*(1+add(binomial(n,j)*polylog(-j,I)*3^j for j in (0..n))))
[int(A225147(n)) for n in (0..19)]

a(n) = Im(-2*i*(1+Sum_{j=0..n}(binomial(n,j)*Li{-j}(i)*3^j))).
For a recurrence see the Maple program.
G.f.: conjecture -T(0)/(1+2*x), where T(k) = 1 - 9*x^2*(k+1)^2/(9*x^2*(k+1)^2 + (1+2*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) = -(-3)^n*skp(n, 2/3), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
G.f.: A225147 = -1/T(0), where T(k) = 1 + 2*x + (k+1)^2*(3*x)^2/ T(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 29 2014
E.g.f.: -exp(-2*x)*sech(3*x). - Sergei N. Gladkovskii, Sep 29 2014
a(n) ~ n! * (sqrt(3)*sin(Pi*n/2) - cos(Pi*n/2)) * 2^(n+1) * 3^n / Pi^(n+1). - Vaclav Kotesovec, Sep 29 2014
From Peter Bala, Nov 13 2016: (Start)
a(n) = - 6^n*E(n,1/6), where E(n,x) denotes the Euler polynomial of order n.
a(2*n) = (-1)^(n+1)*A002438(n); a(2*n+1) = (1/2)*(-1)^n*A002439(n). (End)

A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

Original entry on oeis.org

0, 0, 1, -3, -5, 45, 61, -1113, -1385, 42585, 50521, -2348973, -2702765, 176992725, 199360981, -17487754833, -19391512145, 2195014332465, 2404879675441, -341282303124693, -370371188237525, 64397376340013805, 69348874393137901, -14499110277050234553

Offset: 0

Peter Luschny, Apr 17 2014

sign

			G.f. = x^2 - 3*x^3 - 5*x^4 + 45*x^5 + 61*x^6 - 1113*x^7 - 1385*x^8 + ...

L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.

Cf. A000364, A147315, A186370, A241242.

Cf. A239005, A239322.

A240559 := proc(n) euler(n,1/2) + euler(n,1); if n mod 2 = 1 then % + 2*(euler(n+1,1/2)+euler(n+1,1)) fi; -2^n*% end: seq(A240559(n),n=0..19);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; skp[n_, x0_?NumericQ] := skp[n, x] /. x -> x0; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(skp[k, 0] + skp[k+1, -1]), {k, 0, n}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 09 2014, after Peter Luschny *)

# Efficient computation with L. Seidel's boustrophedon transformation.
def A240559_list(n) :
    A = [0]*(n+1); A[0] = 1; R = [0]
    k = 0; e = 1; x = -1; s = -1
    for i in (0..n):
        Am = 0; A[k + e] = 0; e = -e;
        for j in (0..i): Am += A[k]; A[k] = Am; k += e
        if e == 1: x += 1; s = -s
        v = -A[-x] if e == 1 else A[-x] - A[x]
        if i > 1: R.append(s*v)
    return R
A240559_list(24)

a(2*n) = (-1)^(n+1)*A147315(2*n,1) = (-1)^(n+1)*A186370(2*n,2*n) =(-1)^(n+1)*A000364(n) for n>0.
a(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1) = A241242(n).
a(n) = Sum_{k=0..n} (-1)^(n-k)*2^k*binomial(n,k)*(E(k,1/2) + 2*E(k+1,0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(skp(k,0) + skp(k+1,-1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = A239322(n) + A239005(n+1) - A239005(n). - Paul Curtz, Apr 18 2014
E.g.f.: 1 - sech(x) - tanh(x) + sinh(x)*sech(x)^2 = ((exp(-x)-1)*sech(x))^2 / 2. - Sergei N. Gladkovskii, Nov 20 2014
E.g.f.: (1 - sech(x)) * (1 - tanh(x)). - Michael Somos, Nov 22 2014

A363393 Triangle read by rows. T(n, k) = [x^k] (2 - Sum_{k=0..n} binomial(n, k)Euler(k, 1)(-2*x)^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, -2, 1, 4, 0, -8, 0, 1, 5, 0, -20, 0, 16, 1, 6, 0, -40, 0, 96, 0, 1, 7, 0, -70, 0, 336, 0, -272, 1, 8, 0, -112, 0, 896, 0, -2176, 0, 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936, 1, 10, 0, -240, 0, 4032, 0, -32640, 0, 79360, 0

Offset: 0

Peter Luschny, Jun 04 2023

The Swiss-Knife polynomials (A081658 and A153641) generate the dual triangle ('dual' in the sense of Euler-tangent versus Euler-secant numbers).

			The triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 0;
[3] 1, 3, 0,   -2;
[4] 1, 4, 0,   -8, 0;
[5] 1, 5, 0,  -20, 0,   16;
[6] 1, 6, 0,  -40, 0,   96, 0;
[7] 1, 7, 0,  -70, 0,  336, 0,  -272;
[8] 1, 8, 0, -112, 0,  896, 0, -2176, 0;
[9] 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936;

Peter Luschny, Illustrating the polynomials P363393.
Peter Luschny, Swiss-Knife polynomials and Euler numbers.
Peter Luschny, The Swiss-Knife polynomials.
Index entries for sequences related to Euler numbers.

Cf. A122045 (alternating row sums), A119880 (row sums), A214447 (central column), A155585 (main diagonal), A109573 (subdiagonal), A162660 (variant), A000364.

Cf. A081658, A153641, A220901, A318254, A346838.

P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1) / (n + 1):  T := (n, k) -> coeff(P(n), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
# Second program, based on the generating functions of the columns:
ogf := n -> -(-2)^n * euler(n, 1) / (x - 1)^(n + 1):
ser := n -> series(ogf(n), x, 16):
T := (n, k) -> coeff(ser(k), x, n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative, based on the bivariate generating function:
egf :=  exp(x*y) * (1 + tanh(y)): ord := 20:
sery := series(egf, y, ord): polx := n -> coeff(sery, y, n):
coefx := n -> seq(n! * coeff(polx(n), x, n - k), k = 0..n):
for n from 0 to 9 do coefx(n) od;

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k % 2 == 0:  return 0
    if k == n: return 1 - sum(T(n, j) for j in range(1, n, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 1)])

def B(n: int):
    return bernoulli_polynomial(1, n)
def P(n: int):
    return sum(binomial(n + 1, j) * B(j) * (4^j - 2^j) * x^(j - 1)
           for j in range(n + 2)) / (n + 1)
for n in range(10): print(P(n).list())

For a recursion see the Python program.
T(n, k) = [x^k] P(n, x) where P(n, x) = (1 / (n + 1)) * Sum_{j=0..n+1} binomial(n + 1, j) * Bernoulli(j, 1) * (4^j - 2^j) * x^(j - 1).
Integral_{x=-n..n} P(n, x)/2 dx = n.
T(n, k) = [x^(n - k)] -(-2)^k * Euler(k, 1) / (x - 1)^(k + 1).
T(n, k) = n! * [x^(n - k)][y^n] exp(x*y) * (1 + tanh(y)).

Simpler name by Peter Luschny, Nov 17 2024

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).

A156201 Numerator of Euler(n, 1/8).

Original entry on oeis.org

1, -3, -7, 117, 497, -15123, -95767, 4116837, 34741217, -1921996323, -20273087527, 1370953667157, 17352768515537, -1386843017916723, -20479521468959287, 1888542637550347077, 31872138933891307457, -3331009898404800736323, -63243057486503656319047

Offset: 0

N. J. A. Sloane, Nov 07 2009

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

For denominators see A001018. Cf. A000813.

p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156201 := n -> Im(p(n));
seq(A156201(i), i=0..10);  # Peter Luschny, Apr 29 2013

Table[EulerE[n, 1/8] // Numerator, {n, 0, 18}] (* Jean-François Alcover, Apr 30 2013 *)

a(n) = Im(2*i*(1+Sum_{j=0..n} (binomial(n,j)*Li_{-j}(i)*4^j))). - Peter Luschny, Apr 29 2013
G.f.: conjecture T(0)/(1+3*x), where T(k) = 1 - 16*x^2*(k+1)^2/(16*x^2*(k+1)^2 + (1+3*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) = (-4)^n*skp(n, 3/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014
a(n) = 2^(4*n+1)*(zeta(-n,1/16)-zeta(-n, 9/16)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
From Emanuele Munarini, Aug 22 2022: (Start)
E.g.f.: (2*e^t)/(e^(8*t)+1).
E.g.f. for the sequence of the absolute values: (cos(3*t)+sin(3*t))/cos(4*t).
|a(2*n)| = Sum_{k=0..n} binomial(2*n,2*k) (-1)^k 4^(2*n-2*k) 3^(2*k) |E(2*n-2k)|
|a(2*n+1)| = Sum_{k=0..n} binomial(2*n+1,2*k+1) (-1)^k 4^(2*n-2*k) 3^(2*k+1) |E(2*n-2*k)|
where the E(n)'s are the Euler numbers (A122045).
(End)

A214445 a(n) = Euler(2n)binomial(4n,2n).

Original entry on oeis.org

1, -6, 350, -56364, 17824950, -9334057876, 7308698191340, -7997684730384600, 11655857682806336550, -21824608434847162167300, 51054382673481634917970500, -145916894745749901373155951720, 500306549034304293474784779805500, -2026855002861172152744641068895033544

Offset: 0

Peter Luschny, Jul 18 2012

sign

Central column of the nonzero coefficients of the Swiss-Knife polynomials, a(n) = A153641(4*n, 2*n).

Cf. A086646, A153641.

def A214445(n) : return binomial(4*n,2*n)*euler_number(2*n)

a(n) = (-1)^n * A086646(2n,n). - Alois P. Heinz, Apr 27 2023

A214447 a(n) = (-2)^n * Euler_polynomial(n,1) * binomial(2*n,n).

Original entry on oeis.org

1, -2, 0, 40, 0, -4032, 0, 933504, 0, -385848320, 0, 249576198144, 0, -232643283353600, 0, 295306112919306240, 0, -489743069731226910720, 0, 1028154317960939805081600, 0, -2665182817368374114506506240, 0, 8360422228704533182913131315200

Offset: 0

Peter Luschny, Jul 18 2012

sign

Central column of the Euler tangent triangle, a(n) = A081733(2*n,n).

Also a(n) = -A162660(2*n,n) for n > 0. - Peter Luschny, Jul 23 2012

Cf. A081733, A214445.

A214447 := n -> binomial(2*n,n)*(-2)^n*euler(n,1):
seq(A214447(n), n=0..23);

from mpmath import mp, fac2
mp.dps = 32
def A214447(n) : return (n+1)*2^n*fac2(2*n-1)*add(add((-1)^(k-j)*(k-j)^n / (factorial(j)*factorial(n-j+1)) for j in (0..k)) for k in (1..n))
print([int(A214447(n)) for n in (0..19)]) # Peter Luschny, Jul 23 2012

a(n) = [x^n] (skp(2*n,x+1)-skp(2*n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641.

a(n) = (n+1)*2^n*(2*n-1)!!*Sum_{k=1..n} Sum_{j=0..k} (-1)^(k-j)*(k-j)^n/(j!*(n-j+1)!) for n > 0. - Peter Luschny, Jul 23 2012

A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

Original entry on oeis.org

1, 1, -1, -5, 5, 61, -61, -1385, 1385, 50521, -50521, -2702765, 2702765, 199360981, -199360981, -19391512145, 19391512145, 2404879675441, -2404879675441, -370371188237525, 370371188237525, 69348874393137901, -69348874393137901, -15514534163557086905

Offset: 0

Paul Curtz, Apr 17 2014

sign

A version of the Seidel triangle (1877) for the integer Euler numbers is
              1
           1    1
         2    2   1
       2   4    5   5
    16  16   14  10   5
  16  32  46   56  61  61
etc.
It is not in the OEIS. See A008282.
The first diagonal, Es(n) = 1, 1, 1, 5, 5, 61, 61, 1385, 1385, ..., comes from essentially A000364(n) repeated.
a(n) is Es(n) signed two by two.
Difference table of a(n):
     1,    1,   -1,   -5,     5,    61,   -61, -1385, ...
     0,   -2,   -4,   10,    56,  -122, -1324, ...
    -2,   -2,   14,   46,  -178, -1202, ...
     0,   16,   32, -224, -1024, ...
    16,   16, -256, -800, ...
     0, -272, -544, ...
  -272, -272, ...
     0, ...
etc.
Sum of the antidiagonals: 1, 1, -5, -11, 61, 211, -385, ... = A239322(n+1).
Main diagonal interleaved with the first upper diagonal: 1, 1, -2, -4, 14, 46, ... = signed A214267(n+1).
Inverse binomial transform (first column): A155585(n+1).
The Akiyama-Tanigawa transform applied to A046978(n+1)/A016116(n) gives
    1,   1,   1/2,   0, -1/4, -1/4, -1/8, 0, ...
    0,   1,   3/2,   1,    0, -3/4, -7/8, ...
   -1,  -1,   3/2,   4, 15/4,  3/4, ...
    0,  -5, -15/2,   1,   15, ...
    5,   5, -51/2, -56, ...
    0,  61, 183/2, ...
  -61, -61, ...
    0, ...
etc.
A122045(n) and A239005(n) are reciprocal sequences by their inverse binomial transform. In their respective difference table, two different signed versions of A214247(n) appear: 1) interleaved main diagonal and first under diagonal (1, -1, -1, 2, 4, -14, ...) and 2) interleaved main diagonal and first upper diagonal (1, 1, -1, -2, 4, 14, ...).

G. C. Greubel, Table of n, a(n) for n = 0..475

Cf. A000364, A000657, A008282, A016116, A046978, A119880, A122045, A153641, A155585, A214247, A214267, A239005, A239322.

EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
Euler:= func< n | 2^n*EulerPoly(n, 1/2) >; // A122045
[Euler(n) - Euler(n+1): n in [0..40]]; // G. C. Greubel, Jun 07 2023

A241209 := proc(n) local v, k, h, m; m := `if`(n mod 2 = 0, n, n+1);
h := k -> `if`(k mod 4 = 0, 0, (-1)^iquo(k,4));
(-1)^n*add(2^iquo(-k,2)*h(k+1)*add((-1)^v*binomial(k,v)*(v+1)^m, v=0..k)
,k=0..m) end: seq(A241209(n),n=0..24); # Peter Luschny, Apr 17 2014

skp[n_, x_]:= Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}];
a[n_]:= skp[n, x] - skp[n+1, x]/. x->0; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 17 2014, after Peter Luschny *)
Table[EulerE[n] - EulerE[n+1], {n,0,30}] (* Vincenzo Librandi, Jan 24 2016 *)
-Differences/@Partition[EulerE[Range[0,30]],2,1]//Flatten (* Harvey P. Dale, Apr 16 2019 *)

[euler_number(n) - euler_number(n+1) for n in range(41)] # G. C. Greubel, Jun 07 2023

a(n) = A119880(n+1) - A119880(n).
a(n) is the second column of the fractional array.
a(n) = (-1)^n*second column of the array in A239005(n).
a(n) = skp(n, 0) - skp(n+1, 0), where skp(n, x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 17 2014
E.g.f.: exp(x)/cosh(x)^2. - Sergei N. Gladkovskii, Jan 23 2016
G.f. T(0)/x-1/x, where T(k) = 1 - x*(k+1)/(x*(k+1)-(1-x)/(1-x*(k+1)/(x*(k+1)+(1-x)/T(k+1)))). - Sergei N. Gladkovskii, Jan 23 2016

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

Original entry on oeis.org

0, -3, 45, -1113, 42585, -2348973, 176992725, -17487754833, 2195014332465, -341282303124693, 64397376340013805, -14499110277050234553, 3840151029102915908745, -1182008039799685905580413, 418424709061213506712209285, -168805428822414120140493978273

Offset: 0

Peter Luschny, Apr 17 2014

sign

Cf. A000182, A000364, A028296, A240559, A147315, A186370.

A241242 := proc(n) e := n -> euler(n,1/2) + euler(n,1); -2^(2*n+1)*(e(2*n+1) + 2*e(2*n+2)) end: seq(A241242(n),n=0..15);

Array[-2^(2 # + 1)*(EulerE[2 # + 1, 1/2] + EulerE[2 # + 1, 1] + 2 (EulerE[2 # + 2, 1/2] + EulerE[2 # + 2, 1])) &, 16, 0] (* Michael De Vlieger, May 24 2018 *)

a(n) = A240559(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1).
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*2^k*(E(k, 1/2) + 2*E(k+1, 0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*(skp(k, 0) + skp(k+1, -1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = Bernoulli(2*n + 2) * 4^(n+1) * (1 - 4^(n+1)) / (2*n + 2) - EulerE(2*n + 2), where EulerE(2*n) is A028296. - Daniel Suteu, May 22 2018
a(n) = (-1)^(n+1) * (A000182(n+1) - A000364(n+1)). - Daniel Suteu, Jun 23 2018

A156205 Numerator of Euler(n, 3/8).

Original entry on oeis.org

1, -1, -15, 47, 1185, -6241, -230895, 1704527, 83860545, -796079041, -48942778575, 567864586607, 41893214676705, -574448847467041, -49441928730798255, 782259922208550287, 76946148390480577665, -1379749466246228538241, -152682246738275154625935

Offset: 0

N. J. A. Sloane, Nov 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

For denominators see A001018. Cf. A000813.

p := proc(n) local j; 2*I*(1+add(binomial(n,j)*polylog(-j,I)*4^j, j=0..n)) end:  A156205 := n -> (-1)^(n+1)*Re(p(n));
seq(A156205(i),i=0..11);  # Peter Luschny, Apr 29 2013

Numerator[EulerE[Range[0,20],3/8]] (* Vincenzo Librandi, May 04 2012 *)

a(n) = (-1)^(n+1)*Re(2*I*(1+sum_{j=0..n}(binomial(n,j)*Li_{-j}(I)*4^j))). - Peter Luschny, Apr 29 2013

a(n) = (-4)^n*skp(n, 1/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

cp's OEIS Frontend

A225147 a(n) = Im((1-I)^(1-n)*A_{n, 3}(I)) where A_{n, k}(x) are the generalized Eulerian polynomials.

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A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

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A363393 Triangle read by rows. T(n, k) = [x^k] (2 - Sum_{k=0..n} binomial(n, k)*Euler(k, 1)*(-2*x)^k).

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

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A156201 Numerator of Euler(n, 1/8).

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A214445 a(n) = Euler(2*n)*binomial(4*n,2*n).

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A214447 a(n) = (-2)^n * Euler_polynomial(n,1) * binomial(2*n,n).

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A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

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A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

A363393 Triangle read by rows. T(n, k) = [x^k] (2 - Sum_{k=0..n} binomial(n, k)Euler(k, 1)(-2*x)^k).

A214445 a(n) = Euler(2n)binomial(4n,2n).

A241242 a(n) = -2^(2n+1)(E(2n+1, 1/2) + E(2n+1, 1) + 2(E(2n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.