A188514 Expansion of exp( Sum_{n >= 1} A188458(n)*x^n/n ).

1, 1, -1, -5, 11, 91, -391, -4115, 27971, 357331, -3353731, -50789375, 607914581, 10692083221, -155442170521, -3120028100285, 53341649623091, 1204301220497011, -23663734574555011, -593828627529030095, 13182525824990398001

Offset

0,4

Comments

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

The e.g.f. of this sequence is the product of the e.g.f. of A188458 and an even function (see formula section).

From Peter Bala, Mar 10 2015: (Start)

Note exp( Sum_{n >= 1} A212435(n)*x^n/n ) = exp( -x - 3*x^2/2 + 11*x^3/3 + 57*x^4/4 - ... ) = 1 - x - x^2 + 5*x^3 + 11*x^4 - 91*x^5 - 391*x^6 + + - - ... appears to give this sequence but with a different pattern of signs.

More generallly, it appears that when h is an integer and k is a nonzero integer, the expansion of exp( Sum_{n >= 1} (4*k)^n*E(n,h/(4*k))*x^n/n ) has integer coefficients, where E(n,x) denotes the n-th Euler polynomial. (End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..200

Formula

G.f.: A(x) = 1/(1-x/(1+2*x/(1 -3*x/(1+3*x/(1+x -5*x/(1+5*x/(1+x -7*x/(1+7*x/(1+x -9*x/(1+9*x/(1+x -11*x/(1+11*x/(1+x -... ))))))))))))) (continued fraction).

Let E(x) be the e.g.f. of this sequence, and let G(x) be the e.g.f of A092635 such that G(x) = G(-x)*exp(-4*x), then E(x) and G(x) are related by:

(1) E(x) = exp(-x) * G(-x),

(2) E(x) = exp(x)/cosh(2*x) * (G(x)+G(-x))/2.

Example

O.g.f.: A(x) = 1 + x - x^2 - 5*x^3 + 11*x^4 + 91*x^5 - 391*x^6 +...
Illustration of the properties of the exponential generating function.
E.g.f.: E(x) = 1 + x - x^2/2! - 5*x^3/3! + 11*x^4/4! + 91*x^5/5! - 391*x^6/6! +...
Note that E(x)*cosh(2*x)/exp(x) is an even function:
E(x)*cosh(2*x)/exp(x) = 1 + 2*x^2/2! - 10*x^4/4! + 212*x^6/6! - 10330*x^8/8! + 926972*x^10/10! +...+ A092635(2*n)*x^(2*n)/(2*n)! +...
which equals (G(x)+G(-x))/2 with G(x) being the e.g.f of A092635:
G(x) = 1 - 2*x + 2*x^2/2! + 4*x^3/3! - 10*x^4/4! - 92*x^5/5! + 212*x^6/6! +...

Maple

exp(add(4^n*euler(n, 3/4)*x^n/n, n = 1 .. 20)): seq(coeftayl(%, x = 0, n), n = 0 .. 20); # Peter Bala, Mar 09 2015

Mathematica

A188458:= With[{nn = 160}, CoefficientList[Series[E^x/Cosh[2*x], {x, 0, nn}], x]*Range[0, nn]!]; a:= With[{nmax = 80}, CoefficientList[ Series[Exp[Sum[A188458[[k + 1]]*x^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 51}] (* G. C. Greubel, Aug 26 2018 *)

Prog

(PARI) {A188458(n)=local(X=x+x*O(x^n)); n!*polcoeff(exp(X)/cosh(2*X), n)}
{a(n)=polcoeff(exp(sum(m=1, n, A188458(m)*x^m/m)+x*O(x^n)), n)}
(PARI) {A092635(n)=if(n<0, 0, polcoeff(exp(intformal(serlaplace(-1/cosh(x*2+x*O(x^n))^2*2))), n))} /* Michael Somos */
{a(n)=n!*polcoeff(exp(-x+x*O(x^n))*sum(m=0, n, A092635(m)*(-x)^m/m!), n)}

Crossrefs

Cf. A188458, A092635.

Cf. A212435, A255883.

Sequence in context: A228503 A128454 A368013 * A301923 A353890 A120778

Adjacent sequences: A188511 A188512 A188513 * A188515 A188516 A188517

Keyword

sign

Author

Paul D. Hanna, Apr 02 2011

Status

approved