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 A188458 Expansion of e.g.f. exp(x)/cosh(2*x). 9
 1, 1, -3, -11, 57, 361, -2763, -24611, 250737, 2873041, -36581523, -512343611, 7828053417, 129570724921, -2309644635483, -44110959165011, 898621108880097, 19450718635716001, -445777636063460643, -10784052561125704811, 274613643571568682777 (list; graph; refs; listen; history; text; internal format)
 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: A052442 A217034 A180112 * A212435 A001586 A126201 Adjacent sequences:  A188455 A188456 A188457 * A188459 A188460 A188461 KEYWORD sign AUTHOR Paul D. Hanna, Apr 01 2011 STATUS approved

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Last modified October 23 19:54 EDT 2019. Contains 328373 sequences. (Running on oeis4.)