A188458 Expansion of e.g.f. exp(x)/cosh(2*x).

1, 1, -3, -11, 57, 361, -2763, -24611, 250737, 2873041, -36581523, -512343611, 7828053417, 129570724921, -2309644635483, -44110959165011, 898621108880097, 19450718635716001, -445777636063460643, -10784052561125704811, 274613643571568682777

Offset

0,3

Comments

A signed version of A001586 (Springer numbers).

Equals the logarithmic derivative of A188514 (ignoring the initial term of this sequence); note that the unsigned version (A001586) does not form a logarithmic derivative of an integer sequence.

Links

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

Formula

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

L.g.f.: log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + (-1)^n*x)^n.

E.g.f.: 1 = Sum_{n>=0} a(n)*exp(-(-1)^n*x)*x^n/n!.

G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + (-1)^n*x)^(n+1).

G.f.: 1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + (-1)^n*x)^(n+m) for m>=1.

a(n) = Sum_{k=0..n} 2^k C(n,k) Euler(k). - Peter Luschny

a(n) = (-1)^[n/2]*((1+I)/2)^n * Sum_{k=0..n} ((1-I)/(1+I))^k * Sum_{j=0..k} (-1)^(k-j)*C(n+1, k-j)*(2*j+1)^n. - Peter Bala

O.g.f.: 1/(1-x/(1+4*x/(1-x- 4*x/(1+4*x/(1+x- 6*x/(1+6*x/(1+x- 8*x/(1+8*x/(1+x- 10*x/(1+10*x/(1+x- 12*x/(1+12*x/(1+x- ...))))))))))))) (continued fraction).

E.g.f.: E(x) = exp(x)/cosh(2*x) = 2/G(0) where G(k)= 1 -((-1)^k)*3^k/(1 - x/(x + (k+1)*((-1)^k)*3^k/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jun 07 2012

a(n) ~ n! * (cos(n*Pi/2) + sin(n*Pi/2)) * 2^(2*n+3/2) / Pi^(n+1). - Vaclav Kotesovec, Oct 07 2013

G.f.: conjecture T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 + (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013

From Peter Luschny, Apr 19 2014: (Start)

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

a(n) = 4^n*E(n, 3/4), where E(n,x) are Euler polynomials.

a(n) = (8^n/((n+1)/2))*(B(n+1, 7/8) - B(n+1, 3/8)), where B(n,x) are the Bernoulli polynomials. (End)

a(n) = 2^(3*n+1)*(Zeta(-n,3/8)-Zeta(-n,7/8)). - Peter Luschny, Oct 15 2015

Example

E.g.f.: exp(x)/cosh(2*x) = 1 + x - 3*x^2/2! - 11*x^3/3! + 57*x^4/4! + 361*x^5/5! +...
Illustration of other generating functions.
E.g.f.: 1 = exp(-x) + exp(x)*x - 3*exp(-x)*x^2/2! - 11*exp(x)*x^3/3! +...
L.g.f.: log(1+x) = x/(1-x) - 3*(x^2/2)/(1+x)^2 - 11*(x^3/3)/(1-x)^3 +...
G.f.: 1 = 1/(1+x) + 1*x/(1-x)^2 - 3*x^2/(1+x)^3 - 11*x^3/(1-x)^4 +...
G.f.: 1 = 1/(1+x)^2 + 1*2*x/(1-x)^3 - 3*3*x^2/(1+x)^4 - 11*4*x^3/(1-x)^5 +...
G.f.: 1 = 1/(1+x)^3 + 1*3*x/(1-x)^4 - 3*6*x^2/(1+x)^5 - 11*10*x^3/(1-x)^6 +...

Maple

seq(4^n*euler(n, 3/4), n=0..20); # Peter Luschny, Apr 19 2014

Mathematica

CoefficientList[Series[E^x/Cosh[2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(X)/cosh(2*X), n)}
(PARI) {a(n)=n!*polcoeff(1-sum(k=0, n-1, a(k)*exp(-(-1)^k*x+x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+(-1)^k*x+x*O(x^n))^(k+1)), n)}
(PARI) /* Holds for m>=1: */
{a(n)=local(m=1); polcoeff(1-sum(k=0, n-1, a(k)*binomial(m+k-1, k)*x^k/(1+(-1)^k*x+x*O(x^n))^(k+m)), n)/binomial(m+n-1, n)}
(PARI) /* Recurrence: */
{a(n)=if(n<0, 0, if(n==0, 1, sum(k=1, n, -(-1)^(n*k)*binomial(n, k)*a(n-k))))}
(PARI) {EULER(n)=n!*polcoeff(1/cosh(x+x*O(x^n)), n)}
{a(n)=sum(k=0, n, 2^k*binomial(n, k)*EULER(k))}
(PARI) {a(n)=(-1)^(n\2)*((1+I)/2)^n*sum(k=0, n, ((1-I)/(1+I))^k*sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*(2*j+1)^n))}

Crossrefs

Cf. A001586, A188514.

Sequence in context: A217034 A330351 A180112 * A212435 A001586 A126201

Adjacent sequences: A188455 A188456 A188457 * A188459 A188460 A188461

Keyword

sign

Author

Paul D. Hanna, Apr 01 2011

Status

approved