A002425 -id:A002425 - cp's OEIS frontend

A131758 Coefficients of numerators of rational functions whose binomial transforms give the normalized polylogarithms Li(-n,t)/n!.

1, 0, 1, -1, 1, 2, 4, -14, 10, 6, -15, 83, -157, 89, 24, 56, -424, 1266, -1724, 826, 120, -185, 1887, -8038, 17642, -19593, 8287, 720, 204, -4976, 36226, -126944, 239576, -234688, 90602, 5040

Offset: 0

Tom Copeland, Sep 17 2007, Sep 27 2007, Sep 30 2007, Oct 01 2007, Oct 08 2007

Coefficients may be generated from a modified Riordan array (MRA) formed from Rgf(z,t) = (t/(1+z))/(exp(-z/(1+z))-t) with each row of the array acting to generate the succeeding polynomial P(n,t) from the preceding n polynomials.
The MRA is constructed by appending an n! to the left of the n-th row of the Riordan array A129652 and removing the unit diagonal.
The MRA is partially
    1;
    1,    1;
    2,    3,    2;
    6,   13,    9,    3;
   24,   73,   52,   18,    4;
  120,  501,  365,  130,   30,    5;
  720, 4051, 3006, 1095,  260,   45,   6;
For the MRA:
  1) First column is the n!'s.
  2) Second column is A000262.
Then, e.g., from the terms in the MRA,
  P(0,t) = 0!*(t-1)^0 = 1 from the n=0 row,
  P(1,t) = 1!*(t-1)^1 + 1*P(0,t) = t from the n=1 row,
  P(2,t) = 2!*(t-1)^2 + 3*P(0,t)*(t-1)^1 + 2*P(1,t)
  P(3,t) = 3!*(t-1)^3 + 13*P(0,t)*(t-1)^2 + 9*P(1,t)*(t-1)^1 + 3*P(2,t)
generating
  P(0,t) = (1)
  P(1,t) = (0, 1)
  P(2,t) = (-1, 1, 2)
  P(3,t) = (4, -14, 10, 6) = 4 + -14 t + 10 t^2 + 6 t^3
  P(4,t) = (-15, 83, -157, 89, 24)
  P(5,t) = (56, -424, 1266, -1724, 826, 120)
  P(6,t) = (-185, 1887, -8038, 17642, -19593, 8287, 720)
  P(7,t) = (204, -4976, 36226, -126944, 239576, -234688, 90602, 5040)
For the polynomial array:
  1) The first column is A009940 = (-1)^n * n!*Lag(n,1) =(-1)^n* n!* Lag(n,-1,-1).
  2) Row sums are n!.
  3) Highest order coefficient is n!.
  4) Alternating row sum is below.
Then, with Rf(n,t) = [ t/(1-t)^(n+1) ] * P(n,t)/n!, the polylogs are given umbrally by
  Li(-n,t)/n! = [ 1 + Rf(.,t) ]^n for n = 0,1,2,... so conversely
  Rf(n,t) = {[ Li(-(.),t))/(.)! ]-1}^n.
Note umbrally [ Rf(.,t) ]^n = Rf(n,t) and
  (1+Rf)^0 = 1^0 * [ Rf(.,t) ]^0 = Rf(0,t) = t/(1-t) = Li(0,t).
More generally, Newton interpolation holds and for Re(s) >= 0,
  Li(-s,t)/(s)! = [ 1 + Rf(.,t) ]^s, when convergent in t.
Alternatively, the Rf's may be formed through differentiation of their o.g.f., the Rgf(z,t) above, which may also be written as
  Rgf(z,t) = Sum_{k>=1} [ t^k ] * exp[ k * z/(z+1) ]/(z+1)
    = Sum_{n>=0} [ (-z)^n ] * Sum_{k>=1} [ (t^k * Lag(n,k) ]
    = Sum_{k>=0} [ (-z)^k ] * Lag(k,Li(-(.),t))
    = Sum_{k>=0} [ z^k ] * {[ Li(-(.),t))/(.)! ]-1}^k
    = exp[ Li(-(.),t)*z/(1+z) ]/(1+z),
and operationally as
  Rgf(z,t) = {Sum_{k>=0} (-z)^k * Lag(k,tD)} [ t/(1-t) ]
    = {Sum_{k>=0} (-z)^k * Lag(k,T(.,:tD:))} [ t/(1-t) ]
    = {Sum_{k>=0} (-z)^k * Sum_{j>=0} Lag(k,j) (tD)^j /j!} [ x/(1-x) ]
where D is w.r.t. x at 0
    = {Sum_{k>=0} (-z)^k*Sum_{j=0..k} (-1)^j*[ 1-Lag(k,.) ]^j*(:tD:)^j/ j!} [ t/(1-t) ]
    = {Sum_{k>=0} (-z)^k * exp[ -[ 1-Lag(k,.) ]* :tD: ]} [ t/(1-t) ]
where (:tD:)^n = t^n * D^n, D is the derivative w.r.t. t unless otherwise stated, Lag(n,x) is a Laguerre polynomial and T(n,t) is a Touchard / Bell / exponential polynomial.
Hence [ t/(1-t)^(n+1) ] * P(n,t)/n! = Rf(n,t)
  = {Sum_{k=0..n} (-1)^n-k)*[ C(m,k)/k! ]*(tD)^k} [ t/(1-t) ]
  = {Sum_{k=0..n} (-1)^(n-k)*[ C(m,k)/k! ]*Sum_{j=0..k} S2(k,j)*(:tD:)^j} [ t/(1-t) ]
  = {Sum_{k>=0} (-1)^(n-k) * Lag(n,k) * (tD)^k/k!} [ x/(1-x) ] where D is w.r.t. x at 0
  = {Sum_{k=0..n} (-1)^(n-k)* [ 1-Lag(n,.) ]^k *(:tD:)^k/k!}[ t/(1-t) ],
where S2(k,j) are the Stirling numbers of the second kind and C(m,k), binomial coefficients.
The P(n,t) are related to the Laguerre polynomials through
  P(n,t) = (-1)^n n! [ (1-t)^(n+1)} ] Sum_{k>=0} [ (t^k*Lag(n,k+1) ] = Sum_{m=0..n} a(n,m) * t^m
where a(n,m) = (-1)^n n! [ Sum_{k=0..m} (-1)^k * C(n+1,k) *Lag(n,m-k+1) ] .
Conjecture for the polynomial array:
The greatest common divisor of the coefficients of each polynomial is given by A060872(n)/n or, equivalently, by A038548(n).
Some e.g.f.'s for the Rf's are
  exp[ -Rf(.,t)*z ] = exp{[ 1-Li(-(.),t)/(.)! ]*z}
    = Sum_{n>=0} { (z^n/n!) * Sum_{k>=1} [ t^k * Lag(n,k) ] }
    = Sum_{k>=1} { t^k * (e^z) * J_0[ 2*sqrt(k*z)}
    = Sum_{n>=0} {(-1)^n*(z^n/n!)*(z^/j!)*Lag(n,-1)*Sum_{k>=1} [ t^k*k^n*(k+1)^j ]}
where J_0(x) is the zeroth Bessel function of the first kind.
The expressions (:tD:)^j}[ t/(1-t) ] and the Laguerre polynomials are intimately connected to Lah numbers and rook polynomials.
Some interesting relations to physics, probability and number theory are, for abs(t) < 1 and abs(z) < 1 at least,
BE(t,z) = Sum_{k>=0} [ (-z)^k ] *[ 1 + Rf(.,t) ]^k
  = Rgf(-z/(1+z),t)/(1+z) = t/{exp(z)-t}, a Bose-Einstein distribution,
FD(t,z) = Sum_{k>=0} [ (-z)^k+1 ] *[ 1 + Rf(.,-t) ]^k
  = -Rgf(-z/(1+z),-t)/(1+z) = t/{exp(z)+t}, a Fermi-Dirac distribution
and as t tends to 1 from below, z*BE(t,z) tends to the Bernoulli e.g.f., which is related by the Mellin transform to (s-1)!*Zeta(s). Taking Mellin transforms of BE and FD w.r.t. z gives the polylogarithm over different domains.
Since BE(2,z) is essentially the e.g.f. for the ordered Bell numbers, it follows that umbrally
  n! * Lag(n,OB(.)) = P(n,2) and
  n! * Lag(n,P(.,2)) = OB(n)
where OB(n) are the signed ordered Bell/Fubini numbers A000670.
I.e., P(n,2) and the ordered Bell numbers form a reciprocal Laguerre combinatorial transform pair,
or, equivalently, P(n,2)/n! and OB(n)/n! form a reciprocal finite difference pair, so
  P(n,2)/n! = (-1)^(n+1) * Rf(n,2) = -{1-[ Li(-(.),2))/(.)! ]}^n and
  OB(n) = -Li(-n,2).
Note that n!*Lag(n,(.)!*Lag(.,x)) = x^n is a true identity for general Laguerre polynomials Lag(n,x,a) with a = -1,0,1,..., so one could look at analogous higher-order reciprocal pairs containing OB(n).
In addition, a mixed-order iterated Laguerre transform gives
  n!*Lag{n,(.)!*Lag[ .,P(.,2),0 ],-1}
    = P(n,2) - n*P(n-1,2)
    = n!*Lag[ n,OB(.),-1 ] = A084358(n), lists of sets of lists.
For Eulerian polynomials, E(n,t), given by A173018 (A008292),
  E(n,t)/n! = [ 1-t+P(.,t)/(.)! ]^n
  P(n,t)/n! = [ E(.,t)/(.)!-(1-t) ]^n, or equivalently
  [ E(.,t)/(1-t) ]^n = n!*Lag[ n,-P(.,t)/(1-t) ]
  [ -P(.,t)/(1-t) ]^n = n!*Lag[ n,E(.,t)/(1-t) ], a Laguerre transform pair.
Then from known relations for the Eulerian polynomials, the alternating row sum of the polynomial array is
  P(n,-1) = (-2)^(n+1) * n! * Lag[ n,c(.)*Zeta(-(.)) ]
where c(n) = [ 2^(n+1) - 1 ] and Zeta is the Riemann zeta function. And so
  Zeta(-n) = n! * Lag[ n,-P(.,-1)/2 ] / [ 2 - 2^(n+2) ],
which also holds, with the summation limit of Lag extended to infinity, for n = s, any complex number with Re(s) > 0.
Then from standard formulas 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) and the binomial C(x,y) = x!/[ (x-y)!*y! ]
2^(n+1) * (1-2^(n+1)) * (-1)^n * Zeta(-n)
  = 2^(n+1) * (1-2^(n+1)) * Ber(n+1)/(n+1)
  = [ -(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 ]
  = (-2)^n * n! * C{T[ .,P(.,-1)/2 ] + n, n}
  = an integer = Q(n)
These are related to the zag numbers A000182 by Zag(n) = abs[ Q(2*n-1) ]. And, abs[ Q(2*n-1) ] / 2^q(n) = Zag(n) / 2^q(n) = A002425(n) with q(n) = A101921.
These may be generalized by letting n = s, a complex number, (or interpolating) to obtain generalized Laguerre functions or confluent hypergeometric functions of the first kind, M(a,b,x), or second kind, U(a,b,x), whose arguments are P(.,-1)/2, such as
E(s,-1)/[ 2^s*s! ] = -2*Li(-s,-1)/s! = (2-2^(s+2)) * Zeta(-s)/s!
  = C{T[ .,P(.,-1)/2 ] + s, s} = Lag[ s,-P(.,-1)/2 ] = M[ -s,1,-P(.,-1)/2 ] or,
GN(s+1)/(s+1)! = EP(s,0)/s! = C{-T[ .,P(.,-1)/2 ]-1, n} = U[ -s,1,-P(.,-1)/2 ]/(s)!
And even more generally
  E(s,t)/(1-t)^s = [ (1-t)/t ] Li(-s,t) = s!*Lag[ s,-P(.,t)/(1-t) ]
  = s! * C{T[ .,P(.,t)/(1-t) ] + s, s} = s! * M[ -s,1,-P(.,t)/(1-t) ] .
The Laguerre polynomial expressions are fundamental as they can be interpolated to form general M[ a,b,-P(.,t)/(1-t) ] or U[ a,b,-P(.,t)/(1-t) ] which can then be related either directly or by binomial transforms to many important Sheffer sequences, not to mention group theory and Riemann surfaces.
Note for frequently occurring expressions above: The Laguerre polynomials of order -1 and 0 are intimately connected to Lah numbers and rook polynomials and (tD)^n [t/(1-t)] = T(n,:tD:) [t/(1-t)] generates an Eulerian polynomial in the numerator of a rational function. - Tom Copeland, Sep 09 2008
The deformed Todd operator on p. 12 of Gunnells and Villegas is Td(a,D) = -D / (a*exp(-D) - 1) = [-D/(1-D)] * Rgf(D/(1-D), 1/a) = -D * BE(1/a,-D) = D * FD(-1/a,-D), where BE and FD are the Bose-Einstein and Fermi-Dirac distributions given above. See also connections among the Eulerian polynomials, Ehrhart polynomials, and the Todd operator in Beck and Robins, especially pages 31 and 37. - Tom Copeland, Jun 20 2017

M. Beck and S. Robins, Computing the Continuous Discretely, illustrated by D. Austin, Springer, 2007.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
P. Gunnells and F. Villegas Lattice polytopes, Hecke operators, and the Ehrhart polynomial, arXiv:math/0405573 [math.CO], 2004.

Cf. A133289, A131202.

a[n_, m_] := (-1)^n *n!*Sum[(-1)^k*Binomial[n+1, k]*LaguerreL[n, m-k+1], {k, 0, m}]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 23 2014 *)

a(n,m) = (-1)^n*n!*Sum_{k=0..m} (-1)^k*C(n+1,k)*Lag(n, m-k+1).

A173018 given as reference for Eulerian polynomials and typo in a Laguerre function corrected by Tom Copeland, Oct 02 2014

A291897 Numerator of E(2*n-1,n), where E(n,x) is the Euler polynomial.

Original entry on oeis.org

1, 9, 125, 32977, 971919, 358472059, 47622059953, 137818710619425, 8141400285401267, 9740358918723188381, 3597069206174040366021, 12859671622917809034800123, 3419734700063005545155284375, 8538628250545609672426471056711, 6181704419438256867205044161777369

Offset: 1

Vladimir Shevelev, Sep 22 2017

Conjecture: a(n) is divisible by (2*n-1)^2.
Robert G. Wilson v verified this conjecture up to 5000.
Note that sometimes a(n) is divisible by (2n-1)^3, for example, for n = 1,3,7,9,... when 2*n-1 = 1,5,13,17,... .

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Robert G. Wilson v, Table of n, a(n) for n = 1..215
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Cf. A002425, A006519, A007814, A157805, A292706.

A291897 := n -> euler(2*n-1, n)*2^(padic[ordp](2*n, 2)):
seq(A291897(n), n=1..15); # Peter Luschny, Sep 22 2017

f[n_] := Numerator@ EulerE[2 n - 1, n]; Array[f, 15] (* Robert G. Wilson v, Sep 22 2017 *)
Table[2^IntegerExponent[2n, 2] EulerE[2 n-1, n], {n,1,15}] (* Peter Luschny, Sep 22 2017 *)

a(n) = numerator(subst(eulerpol(2*n-1, 'x), 'x, n)); \\ Michel Marcus, Sep 21 2021

from sympy import euler
def A291897(n): return euler((n<<1)-1,n).p # Chai Wah Wu, Jul 07 2022

a(n) = (E(2*n-1,n) + (-1)^(n-1)*E(2*n-1,0))*A006519(2*n) + A002425(n).
a(n) = 2*(-1)^n*A292706(n)*A006519(2*n) + A002425(n).
a(n) = E(2*n-1, n)*2^A007814(2*n). - Peter Luschny, Sep 22 2017

More terms from Peter J. C. Moses, Sep 22 2017

A081733 Triangle read by rows, T(n,k) = 2^(n-k)*[x^k] Euler_polynomial(n, x), for n >= 0, k >= 0.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 2, 0, -3, 1, 0, 8, 0, -4, 1, -16, 0, 20, 0, -5, 1, 0, -96, 0, 40, 0, -6, 1, 272, 0, -336, 0, 70, 0, -7, 1, 0, 2176, 0, -896, 0, 112, 0, -8, 1, -7936, 0, 9792, 0, -2016, 0, 168, 0, -9, 1, 0, -79360, 0, 32640, 0, -4032, 0, 240, 0, -10, 1, 353792, 0, -436480, 0, 89760, 0, -7392, 0, 330, 0, -11, 1, 0, 4245504, 0

Offset: 0

Wouter Meeussen, Apr 06 2003

Sum of row n equals Euler(n) (in the sense of the non-official version A122045; R. P. Stanley calls A000111 Euler numbers.)

			The coefficient lists of the first 5 Euler polynomials are {1}, {-1/2, 1}, {0, -1, 1}, {1/4, 0, -3/2, 1}, {0, 1, 0, -2, 1}. Multiply by 2^(n-k) to get
   1,
  -1,  1,
   0, -2,  1,
   2,  0, -3,  1,
   0,  8,  0, -4,  1.

Cf. A002425, A058940, A006519.

Cf. A009006, A155585, A000182, A119468.

T := (n,k) -> 2^(n-k)*coeff(euler(n,x),x,k):
T := (n,k) -> 2^(n-k)*binomial(n,k)*euler(n-k,1): # Peter Luschny, Jan 25 2009

Table[2^n (1/2)^(Range[0, n]) CoefficientList[EulerE[n, x], x], {n, 0, 16}]

def A081733(n, k) : return (-2)^(n-k)*binomial(n,k)*euler_polynomial(n-k,1)
# Peter Luschny, Jul 18 2012

T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(0) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
Matrix inverse is A119468 and central column is A214447. - Peter Luschny, Jul 18 2012
Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then T(n,k) = [x^(n-k)]((skp{n}(x-1) - skp{n}(x+1))/2 + x^n). - Peter Luschny, Jul 22 2012
E.g.f.: exp(z*x)*(1-tanh(x)). - Peter Luschny, Aug 01 2012
E.g.f.: [2/(e^(2t)+1)] e^(tx) = e^(P.(x)t), so this is an Appell sequence with lowering operator D = d/dx and raising operator R = x - 2/[e^(-2D)+1], i.e., D P_n(x) = n P_{n-1}(x) and R p_n(x) = P_{n+1}(x). Also, (P.(x)+y)^n = P_n(x+y), umbrally. R = x - 1 - D + 2 D^3/3! + ... contains the e.g.f.(D) mod signs of A009006 and A155585 and signed, aerated A000182, the zag numbers, and P_n(0) are the coefficients (mod signs/shift) of these sequences. The polynomials PI_n(x) of A119468 are the umbral compositional inverses of this sequence, i.e., P_n(PI.(x)) = x^n = PI_n(P.(x)) under umbral composition. Note that 2/[e^(2t)+1] = 2 Sum_{n >= 0} Eta(-n) (-2t)^n/n!], where Eta(s) is the Dirichlet eta function, and b_n = 2 *(-2)^n Eta(-n) = (-1)^n (2^(n+1)-4^(n+1)) Zeta(-n) = (2^(n+1)-4^(n+1)) B(n+1)/(n+1) with Zeta(s), the Riemann zeta function, and B(n), the Bernoulli numbers, so P_n(x) = (b. + x)^n, as an Appell polynomial. - Tom Copeland, Sep 27 2015

Corrected T(0,0) = Euler(0) = 1 (was 0), Peter Luschny, Sep 30 2010

New name from Peter Luschny, Jul 18 2012

A143074 Numerator of Euler(n,2).

Original entry on oeis.org

1, 3, 2, 9, 2, 3, 2, 33, 2, -27, 2, 699, 2, -5457, 2, 929601, 2, -3202287, 2, 221930589, 2, -4722116517, 2, 968383680843, 2, -14717667114147, 2, 2093660879252679, 2, -86125672563201177, 2, 129848163681107302017, 2, -868320396104950823607, 2, 209390615747646519456969

Offset: 0

N. J. A. Sloane, Nov 10 2009

			By the formula, we have a(1) = 2*2 - 1 = 3, a(3) = 2*4 + 1 = 9, a(5) = 2*2 - 1 = 3, a(7) = 2*8 + 17 = 33, a(9) = 2*2 - 31 = -27, etc. - _Vladimir Shevelev_, Sep 04 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.
Vladimir Shevelev, A formula for numerator of Euler(n,k), Wed Sep 06 2017.

For denominators see A006519.

Cf. A002425, A157805, A157808, A157812, A157827, A157835, A157856, A157864, A157875, A157886, A157907.

Numerator[EulerE[Range[0,40],2]] (* Vincenzo Librandi, May 03 2012 *)

For even n, a(n) = 2 - delta(n,0), where delta is the Kronecker symbol;
for n==1 (mod 4), a(n) = 2*A006519(n+1) - A002425((n+1)/2);
for n==3 (mod 4), a(n) = 2*A006519(n+1) + A002425((n+1)/2). - Vladimir Shevelev, Sep 04 2017

A089170 Numerator of 2BernoulliB[2(n+1)](4^(n+1)-1)/(2(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 527, 1, 1, 1, 1, 31, 1, 1, 731, 1, 41, 1, 1, 1, 37, 1333, 17, 1, 1, 1, 31, 1, 1, 1, 17, 73, 1, 1, 1, 43, 1271, 59, 629, 1, 73, 2759, 43, 1, 1, 1, 17, 1, 67, 7519, 1, 31, 89, 1, 289, 1, 29020032511, 1, 10573, 1, 1, 1, 2227, 486029

Offset: 0

Wouter Meeussen, Dec 07 2003

nonn

Ratios of two similar sequences.

This sequence is related to the sequences of the numerators and denominators of the Taylor series for tan(x), i.e., A002430 and A036279, and the similar sequences A160469 and A156769. - Johannes W. Meijer, May 24 2009

Michael De Vlieger, Table of n, a(n) for n = 0..5000

Cf. A002425.
From Johannes W. Meijer, May 24 2009: (Start)
Equals A160469(n+1)/A002430(n+1).
Equals A156769(n+1)/A036279(n+1).
(End)

seq(numer(2*bernoulli(2*n)*(4^n-1)/(2*n))/numer((4^n-1)*bernoulli(2*n)/(2*n)!),n=1..100); # C. Ronaldo

Table[Numerator[2*BernoulliB[2*n]*(4^n -1)/(2*n)]/Numerator[SeriesCoefficient[Series[1/(1+Cosh[x]), {x, 0, 2n}], 2n-2]], {n, 1, 128}]

For n>=0, a(n)=c(n+1) where c(n)=numerator((4^n-1)*B(2*n)/n)/numerator((4^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

A058940 Triangle of coefficients of Euler polynomials rescaled to integers by multiplication with 2^(binary carry sequence (A007814)).

Original entry on oeis.org

1, -1, 2, 0, -1, 1, 1, 0, -6, 4, 0, 1, 0, -2, 1, -1, 0, 5, 0, -5, 2, 0, -3, 0, 5, 0, -3, 1, 17, 0, -84, 0, 70, 0, -28, 8, 0, 17, 0, -28, 0, 14, 0, -4, 1, -31, 0, 153, 0, -126, 0, 42, 0, -9, 2, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 691, 0, -3410, 0, 2805, 0, -924, 0, 165, 0, -22, 4, 0, 2073, 0, -3410, 0, 1683, 0, -396, 0, 55, 0, -6

Offset: 0

Wouter Meeussen, Jan 12 2001

Sums of even rows are A002425, sums of odd rows are 0, first element of even rows is -row sum, first element of row(2^p) is second element of row(1+2^p), LCM of numerators of Euler polynomial coefficients is A007814.

Cf. A004174, A002425, A007814.

A058940_row := proc(n) local i; seq(coeff(euler(n,x)*2^padic[ordp](n+1,2),x,i), i=0..n) end: # Peter Luschny, Nov 26 2010

Flatten[ Table[ CoefficientList[ EulerE[n, x]*2^IntegerExponent[n+1, 2], x], {n, 0, 12}]] (* Jean-François Alcover, Nov 18 2011, after Wouter Meeussen *)

T(n, k) = [x^k] E(n, x)*2^A007814(n+1).

A090681 Expansion of (sec(x/2)^2 + sech(x/2)^2)/2 in powers of x^4.

Original entry on oeis.org

1, 1, 31, 5461, 3202291, 4722116521, 14717667114151, 86125672563201181, 868320396104950823611, 14129659550745551130667441, 352552873457246307069012458671, 12942188000689093683411117827763301, 675618013651758631167025175564066787331, 48743995308245045290420262686473639399176761

Offset: 0

Benoit Cloitre, Dec 18 2003

nonn

			(sec(x/2)^2 + sech(x/2)^2)/2 = 1 + x^4/4! + 31*x^8/8! + 5461*x^12/12! + ...

Stefano Spezia, Table of n, a(n) for n = 0..135

Cf. A001469, A002425, A012670, A012853.

[2*(4^(2*n+1) -1)*BernoulliNumber(4*n+2)/(2*n+1): n in [0..15]]; // G. C. Greubel, Jun 28 2019

a := n->(2*2^(4*n+2)-2)*bernoulli(4*n+2)/(2*n+1): seq(a(n), n = 0 .. 15); # Stefano Spezia, Jun 14 2019

a[n_]:=2*(2^(4*n+2)-1)*BernoulliB[4*n+2]/(2*n+1); Array[a,15,0] (* Stefano Spezia, Jun 14 2019 *)

a(n)=if(n<0,0,n*=4;n!*polcoeff(1/cosh(x/2+x*O(x^n))^2+1/cos(x/2+x*O(x^n))^2,n)/2) /* Michael Somos, Mar 06 2004 */

a(n)=if(n<0,0,n=4*n+2;4*(2^n-1)*bernfrac(n)/n) /* Michael Somos, Mar 06 2004 */

[2*(4^(2*n+1)-1)*bernoulli(4*n+2)/(2*n+1) for n in (0..15)] # G. C. Greubel, Jun 28 2019

a(n) = -G(2n+1)/(2n+1) where G(k) is the k-th Genocchi number of first kind (A001469).
a(n) = A002425(2n+1).
a(n) = A012853(n)/2^(4n+1).
a(n) = abs(A012670(n)/2^(6n+1)).
E.g.f.: (sec(x/2)^2 + sech(x/2)^2)/2 = Sum_{k>=1} a(k)*x^(4k)/(4k)!. - Michael Somos, Mar 06 2004
a(n) == 1 (mod 30). - Michael Somos, Jul 23 2005

A157805 Numerator of Euler(n,3).

Original entry on oeis.org

1, 5, 6, 55, 30, 125, 126, 2015, 510, 2075, 2046, 15685, 8190, 38225, 32766, 118975, 131070, 3726575, 524286, -217736285, 2097150, 4730505125, 8388606, -968249463115, 33554430, 14717801331875, 134217726, -2093659805510855, 536870910, 86125674710684825

Offset: 0

N. J. A. Sloane, Nov 10 2009

			By the formula, we have a(1) = 2*2 + 1 = 5, a(3) = 14*4 - 1 = 55, a(5) = 62*2 + 1 = 125, a(7) = 254*8 - 17 = 2015, a(9) = 1022*2 + 31 = 2075, etc. - _Vladimir Shevelev_, Sep 04 2017

Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.
Vladimir Shevelev, A formula for numerator of Euler(n,k), Wed Sep 06 2017.

For denominators see A006519.

Cf. A002425.

a2425[n_] := (-1)^n/n*(1 - 4^n)*BernoulliB[2*n]*2^IntegerExponent[2*n, 2];
a6519[n_] := 2^IntegerExponent[n, 2];
a[n_] := Switch[Mod[n, 4], 0 | 2, 2^(n + 1) - 2 + KroneckerDelta[n, 0], 1, (2^(n + 1) - 2)*a6519[n + 1] + a2425[(n + 1)/2], 3, (2^(n + 1) - 2)*a6519[n + 1] - a2425[(n + 1)/2]];
Table[a[n], {n, 0, 30}]
(* or *)
Table[EulerE[n, 3] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Jul 14 2018 *)

From Vladimir Shevelev, Sep 04 2017: (Start)
For even n, a(n) = 2^(n+1) - 2 + delta(n,0), where delta is the Kronecker symbol; for n == 1 (mod 4), a(n) = (2^(n+1)-2)*A006519(n+1) + A002425((n+1)/2); for n == 3 (mod 4), a(n) = (2^(n+1)-2)*A006519(n+1) - A002425((n+1)/2).
A generalization: Let N(n,k) denote numerator of Euler(n,k), k >= 1 is an integer. Set u(n,k) = 2*Sum_{1 <= i <= k-1}(-1)^(i-1)*(k-i)^n. Then, for even n, N(n,k) = u(n,k) + (-1)^(k-1)^delta(n,0); for n == 1 (mod 4), N(n,k) = u(n,k)*A006519(n+1) + (-1)^(k-1)*A002425((n+1)/2); for n == 3 (mod 4), N(n,k) = u(n,k)* A006519 (n+1) - (-1)^(k-1)*A002425((n+1)/2). (End)

A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

Original entry on oeis.org

1, 6, 5, 28, 9, 22, 13, 120, 17, 38, 21, 92, 25, 54, 29, 496, 33, 70, 37, 156, 41, 86, 45, 376, 49, 102, 53, 220, 57, 118, 61, 2016, 65, 134, 69, 284, 73, 150, 77, 632, 81, 166, 85, 348, 89, 182, 93, 1520, 97, 198, 101, 412, 105, 214, 109, 888, 113, 230, 117

Offset: 1

Peter Luschny, Dec 01 2012

An asymptotic expansion of the logarithm of the central binomial (A000984) for n>0 is given by log(binomial(2*n,n)) ~ (n*log(16)-log(Pi)-log(n) + sum_{k>=1}((-4)^(-k)*A002425(k)/a(k)*n^(1-2*k)))/2.

An asymptotic expansion of the logarithm of the swinging factorial (A056040) for n>1 is given by log(swing(n)) ~ (n*log(4)-log(Pi)-(-1)^n*(log(n/2) - (1/2)*sum_{k>=1}((-1)^k*A002425(k)/a(k)*n^(1-2*k))))/2.

			log(binomial(2*n,n)) = n*log(4) - (log(n)+log(Pi))/2 - 1/(8*a(1)*n) + 1/(32*a(2)*n^3) - 1/(128*a(3)*n^5) + 17/(512*a(4)*n^7) - 31/(2048*a(5)*n^9) + 691/(8192*a(6)*n^11) + O(1/n^13).
log(swing(n)) = n*log(2) - (1/2)*log(Pi) - (1/4)*(-1)^n*(2*log(n/2) + 1/(a(1)*n) - 1/(a(2)*n^3) + 1/(a(3)*n^5) - 17/(a(4)*n^7) + 31/(a(5)*n^9) - 691/(a(6)*n^11)) + O(1/n^13).

Peter Luschny, Table of n, a(n) for n = 1..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Cf. A006519, A118413, A385054.

Coeff_list := proc(len) local n;
asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
subs(n=1/n,simplify(convert(%,polynom)));
[seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);

max = 60; s = Normal[Series[Log[x/2]/2+LogGamma[x/2+1/2]-LogGamma[x/2+1], {x, Infinity, 2*max}]] /. x -> 1/x; a[n_] := Denominator[4*Coefficient[s, x^(2*n-1), 1]]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Feb 17 2014 *)
a[n_] := Denominator[2*EulerE[2*n-1, 1]/(2*n-1)]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 04 2014, after Peter Luschny *)

a(n) = denominator(2*E(2*n-1, 1)/(2*n-1)) where E(n, x) is the Euler polynomial. - Peter Luschny, Apr 03 2014

Warning: a(n) != (2*n-1)*2^valuation(n, 2). This was mistakenly assumed several times in the past, see A385054. - Peter Luschny, Jun 17 2025

Edited and incorrect entries removed by Georg Fischer and Peter Luschny, Jun 16 2025

A261042 Generating function g(0) where g(k) = 1 - x2(k+1)(k+2)/(x2(k+1)(k+2) - 1/g(k+1)).

Original entry on oeis.org

1, 4, 64, 2176, 126976, 11321344, 1431568384, 243680935936, 53725527801856, 14893509177769984, 5070334006399074304, 2079588119566033616896, 1011390382859091900891136, 575501120339508919401447424, 378784713733072451034702413824, 285539131625477547496925147693056

Offset: 0

Peter Luschny, Aug 08 2015

More generally let G(y) defined by the Taylor expansion of the continued fraction
g(y,k) = 1 - (y*x*(k+1)*(k+2)) / ((y*x*(k+1)*(k+2)) - 1/g(y,k+1)). Then
G(1/2) -> A002105, G(1) -> A000182, G(2) -> A261042, G(4) -> A253165 and G(1/8)(n) *2^(n-1+padic(n,2)) -> A002425.

Cf. A000182, A002105, A002425, A126156 (example section), A253165.

eulerCF := proc(f, len) local g, k; g := 1;
for k from len-2 by -1 to 0 do g := 1 - f(k)/(f(k)-1/g) od;
PolynomialTools:-CoefficientList(convert(series(g, x, len), polynom), x) end:
A261042_list := len -> eulerCF(k -> x*2*(k+1)*(k+2), len): A261042_list(16);
# Alternative:
ser := series(cos(x/sqrt(2))^(-2), x, 32):
seq(2^(2*n)*(2*n)!*coeff(ser, x, 2*n), n = 0..15); # Peter Luschny, Sep 03 2022

fracGen[f_, len_] := Module[{g, k}, g[len] = 1; For[k = len-1, k >= 0, k--, g[k] = 1-f[k]/(f[k]-1/g[k+1])]; CoefficientList[g[0] + O[x]^(len+1), x] ]; A261042list[len_] := fracGen[x*2*(#+1)*(#+2)&, len-1]; A261042list[16] (* Jean-François Alcover, Aug 08 2015, after Peter Luschny *)

def A261042_list(len):
    f = lambda k: x*2*(k+1)*(k+2)
    g = 1
    for k in range(len-2,-1,-1):
        g = (1-f(k)/(f(k)-1/g)).simplify_rational()
    return taylor(g, x, 0, len-1).list()
A261042_list(16)

a(n) = 2^(2*n)*(2*n)!*[x^(2*n)] cos(x/sqrt(2))^(-2). - Peter Luschny, Sep 03 2022

cp's OEIS Frontend

A131758 Coefficients of numerators of rational functions whose binomial transforms give the normalized polylogarithms Li(-n,t)/n!.

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A291897 Numerator of E(2*n-1,n), where E(n,x) is the Euler polynomial.

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A081733 Triangle read by rows, T(n,k) = 2^(n-k)*[x^k] Euler_polynomial(n, x), for n >= 0, k >= 0.

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A143074 Numerator of Euler(n,2).

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A089170 Numerator of 2*BernoulliB[2*(n+1)]*(4^(n+1)-1)/(2*(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

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A058940 Triangle of coefficients of Euler polynomials rescaled to integers by multiplication with 2^(binary carry sequence (A007814)).

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A090681 Expansion of (sec(x/2)^2 + sech(x/2)^2)/2 in powers of x^4.

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A089170 Numerator of 2BernoulliB[2(n+1)](4^(n+1)-1)/(2(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

A261042 Generating function g(0) where g(k) = 1 - x2(k+1)(k+2)/(x2(k+1)(k+2) - 1/g(k+1)).