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A155585 a(n) = 2^n*E(n, 1) where E(n, x) are the Euler polynomials. 25
1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0, -353792, 0, 22368256, 0, -1903757312, 0, 209865342976, 0, -29088885112832, 0, 4951498053124096, 0, -1015423886506852352, 0, 246921480190207983616, 0, -70251601603943959887872, 0, 23119184187809597841473536, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Previous name was: a(n) = Sum_{k=0..n-1} (-1)^(k)*C(n-1,k)*a(n-1-k)*a(k) for n>0 with a(0)=1.

Factorials have a similar recurrence: f(n) = Sum_{k=0..n-1} C(n-1,k)*f(n-1-k)*f(k), n>0.

Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times Sum_{k=0..n} binomial(n,k)*k^q, then A155585(x) = Sum_{k=0..x-1} T(x,k)*(-1)^k. See Mathematica code below. - John M. Campbell, Nov 16 2011

For the difference table and the relation to the Seidel triangle see A239005. - Paul Curtz, Mar 06 2014

From Tom Copeland, Sep 29 2015: (Start)

Let z(t) = 2/(e^(2t)+1) = 1 + tanh(-t) = e.g.f.(-t) for this sequence = 1 - t + 2 t^3/3! - 16 t^5/5! + ... .

dlog(z(t))/dt = -z(-t), so the raising operators that generate Appell polynomials associated with this sequence, A081733, and its reciprocal, A119468, contain z(-d/dx) = e.g.f.(d/dx) as the differential operator component.

dz(t)/dt = z*(z-2), so the assorted relations to a Ricatti equation, the Eulerian numbers A008292, and the Bernoulli numbers in the Rzadkowski link hold.

From Michael Somos's formula below (drawing on the Edwards link), y(t,1)=1 and x(t,1) = (1-e^(2t))/(1+e^(2t)), giving z(t) = 1 + x(t,1). Compare this to the formulas in my list in A008292 (Sep 14 2014) with a=1 and b=-1,

A) A(t,1,-1) = A(t) = -x(t,1) = (e^(2t)-1)/(1+e^(2t)) = tanh(t) = t + -2 t^3/3! + 16 t^5/5! + -272 t^7/7! + ... = e.g.f.(t) - 1 (see A000182 and A000111)

B) Ainv(t) = log((1+t)/(1-t))/2 = tanh^(-1)(t) = t + t^3/3 + t^5/5 + ..., the compositional inverse of A(t)

C) dA/dt = (1-A^2), relating A(t) to a Weierstrass elliptic function

D) ((1-t^2)d/dt)^n t evaluated at t=0, a generator for the sequence A(t)

F) FGL(x,y)= (x+y)/(1+xy) = A(Ainv(x) + Ainv(y)), a related formal group law corresponding to the Lorentz FGL (Lorentz transformation--addition of parallel velocities in special relativity) and the Atiyah-Singer signature and the elliptic curve (1-t^2)s = t^3 in Tate coordinates according to the Lenart and Zainoulline link and the Buchstaber and Bunkova link (pp. 35-37) in A008292.

A133437 maps the reciprocal odd natural numbers through the refined faces of associahedra to a(n).

A145271 links the differential relations to the geometry of flow maps, vector fields, and thereby formal group laws. See Mathworld for links of tanh to other geometries and statistics.

Since the a(n) are related to normalized values of the Bernoulli numbers and the Riemann zeta and Dirichlet eta functions, there are links to Witten's work on volumes of manifolds in two-dimensional quantum guage theories and the Kervaire-Milnor formula for homotopy groups of hyperspheres (see my link below).

See A101343, A111593 and A059419 for this and the related generator (1 + t^2) d/dt and associated polynomials. (End)

LINKS

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

Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera

T. Copeland, The Kervaire-Milnor formula

H. M. Edwards, A normal form for elliptic curves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 393-422. see section 12, pp. 410-411.

G. Rzadkowski, Bernoulli numbers and solitons-revisited,  Jrn. Nonlinear Math. Physics, 1711, pp. 121-126.

FORMULA

E.g.f.: exp(x)*sech(x) = exp(x)/cosh(x). (See A009006.) - Paul Barry, Mar 15 2006

Sequence of absolute values is A009006 (e.g.f. 1+tan(x)).

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 2*k*x). - Paul D. Hanna, Jul 20 2011

a(n) = 2^n*E_{n}(1) where E_{n}(x) are the Euler polynomials. - Peter Luschny, Jan 26 2009

a(n) = EL_{n}(-1) where EL_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010

a(n) = (4^n-2^n)*B_n(1)/n, where B_{n}(x) are the Bernoulli polynomials (B_n(1) = B_n for n <> 1). - Peter Luschny, Apr 22 2009

G.f.: 1/(1-x+x^2/(1-x+4*x^2/(1-x+9*x^2/(1-x+16*x^2/(1-...))))) (continued fraction). - Paul Barry, Mar 30 2010

G.f.: -log(x/(exp(x)-1))/x = sum(n>=0, a(n)*x^n/(2^(n+1)*(2^(n+1)-1)*n!). - Vladimir Kruchinin, Nov 05 2011

E.g.f.: exp(x)/cosh(x) = 2/(1+exp(-2*x)) = 2/(G(0) + 1) ; G(k) = 1 - 2*x/(2*k + 1 - x*(2*k+1)/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2011

E.g.f. is x(t,1) + y(t,1) where x(t,a) and y(t,a) satisfy y(t,a)^2 = (a^2 - x(t,a)^2) / (1 - a^2 * x(t,a)^2) and dx(t,a) / dt = y(t,a) * (1 - a * x(t,a)^2) and are the elliptic functions of Edwards. - Michael Somos, Jan 16 2012

E.g.f.: 1/(1 - x/(1+x/(1 - x/(3+x/(1 - x/(5+x/(1 - x/(7+x/(1 - x/(9+x/(1 +...))))))))))), a continued fraction. - Paul D. Hanna, Feb 11 2012

E.g.f. satisfies: A(x) = Sum_{n>=0} Integral( A(-x) dx )^n / n!. - Paul D. Hanna, Nov 25 2013

a(n) = -2^(n+1)*Li_{-n}(-1). - Peter Luschny, Jun 28 2012

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

G.f.: 1 + x/T(0) where T(k) = 1 + (k+1)*(k+2)*x^2/T(k+1)); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012

E.g.f.: exp(x)/cosh(x)= 1 + x/S(0) where S(k)= (2*k+1) + x^2/S(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012

E.g.f.: 1+x/(U(0)+x) where U(k) = 4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1))));(continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012

E.g.f.: 1 + tanh(x) = 4*x/(G(0)+2*x) where G(k) = 1 - (k+1)/(1 - 2*x/(2*x + (k+1)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012

G.f.: 1 / (1 - b(1)*x / (1 - b(2)*x / (1 - b(3)*x / ... ))) where b = A001057. - Michael Somos, Jan 03 2013

G.f.: 1 + x/G(0) where G(k) = 1 + 2*x^2*(2*k+1)^2 - x^4*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1); (continued fraction due to Stieltjes). - Sergei N. Gladkovskii, Jan 13 2013

E.g.f.: 1 + x/( G(0) + x ) where G(k) = 1 - 2*x/(1 + (k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013

G.f.: 2 - 1/Q(0) where Q(k) = 1 + x*(k+1)/( 1 - x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Apr 02 2013

G.f.: 2 - 1/Q(0), where Q(k)= 1 + x*k^2 + x/(1 - x*(k+1)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013

G.f.: 1/Q(0), where Q(k)= 1 - 2*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013

G.f.: 1/Q(0), where Q(k)= 1 - x*(k+1)/(1 + x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

E.g.f.: 1 + x*Q(0), where Q(k) = 1 - x^2/( x^2 + (2*k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013

G.f.: 2 - T(0)/(1+x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1+x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2013

E.g.f.: 1/(x - Q(0)), where Q(k) = 4*k^2 - 1 + 2*x + x^2*(2*k-1)*(2*k+3)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013

From Paul Curtz, Mar 06 2014: (Start)

a(2n) = A000007(n).

a(2n+1) = (-1)^n*A000182(n+1).

a(n) is the binomial transform of A122045(n).

a(n) is the row sum of A081658. For fractional Euler numbers see A238800.

a(n) + A122045(n) = 2, 1, -1, -2, 5, 16,... = -A163982(n).

a(n) - A122045(n) = -A163747(n).

a(n) is the Akiyama-Tanigawa transform applied to 1, 0, -1/2, -1/2, -1/4, 0,... = A046978(n+3)/A016116(n). (End)

a(n) = 2^(2*n+1)*(zeta(-n,1/2)-zeta(-n, 1)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015

a(n)= 2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1, 1]/(n + 1) . (From Bill Gosper, Oct 28 2015) - N. J. A. Sloane, Oct 28 2015

EXAMPLE

E.g.f.: 1 + x - 2*x^3/3! + 16*x^5/5! - 272*x^7/7! + 7936*x^9/9! -+ ... = exp(x)/cosh(x).

O.g.f.: 1 + x - 2*x^3 + 16*x^5 - 272*x^7 + 7936*x^9 - 353792*x^11 +- ...

O.g.f.: 1 + x/(1+2*x) + 2!*x^2/((1+2*x)*(1+4*x)) + 3!*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + ...

MAPLE

A155585 := n -> 2^n*euler(n, 1): # Peter Luschny, Jan 26 2009

MATHEMATICA

a[m_] := Sum[(-2)^(m - k) k! StirlingS2[m, k], {k, 0, m}] (* Peter Luschny, Apr 29 2009 *)

poly[q_] :=  2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] :=  First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Table[Sum[T[x, k]*(-1)^k, {k, 0, x-1}], {x, 1, 16}] (* John M. Campbell, Nov 16 2011 *)

f[n_] := (-1)^n 2^(n+1) PolyLog[-n, -1]; f[0] = -f[0]; Array[f, 27, 0] (* Robert G. Wilson v, Jun 28 2012 *)

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(k)*binomial(n-1, k)*a(n-1-k)*a(k)))}

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(X)/cosh(X), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)} \\ Paul D. Hanna, Jul 20 2011

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( 1 + sinh(x + A) / cosh(x + A), n))} /* Michael Somos, Jan 16 2012 */

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, intformal(subst(A, x, -x)+x*O(x^n))^k/k!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Nov 25 2013

for(n=0, 30, print1(a(n), ", "))

(Sage)

def A155585(n) :

    if n == 0 : return 1

    return add(add((-1)^(j+1)*binomial(n+1, k-j)*j^n for j in (0..k)) for k in (1..n))

[A155585(n) for n in (0..26)] # Peter Luschny, Jul 23 2012

def A155585_list(n) :  Akiyama-Tanigawa algorithm

    A = [0]*(n+1); R = []

    for m in range(n+1) :

        d = divmod(m+3, 4)

        A[m] = 0 if d[1] == 0 else (-1)^d[0]/2^(m//2)

        for j in range(m, 0, -1) :

            A[j - 1] = j * (A[j - 1] - A[j])

        R.append(A[0])

    return R

A155585_list(30)  # Peter Luschny, Mar 09 2014

CROSSREFS

Cf. A001057, A009006, A102573.

Equals row sums of A119879. - Johannes W. Meijer, Apr 20 2011

Cf. A081733, A119468, A008292, A000182, A000111, A101343, A111593, A059419, A133437, A145271.

Sequence in context: A146558 A025600 A009006 * A236755 A057375 A009045

Adjacent sequences:  A155582 A155583 A155584 * A155586 A155587 A155588

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 24 2009

EXTENSIONS

New name by Peter Luschny, Mar 12 2015

STATUS

approved

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Last modified June 1 01:18 EDT 2016. Contains 273548 sequences.