A217066 E.g.f. is series reversion of x*(sec(x)+tan(x)).

1, -2, 9, -68, 725, -9966, 167629, -3334120, 76543785, -1992009850, 57948521521, -1863394764780, 65631109286717, -2512768138160294, 103905545328667125, -4615035074291158352, 219122841820491458897, -11075488610594107402098, 593746153204862664363481

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = sum(k=1..n-1, (-1)^k*binomial(n+k-1,n-1)*sum(k=1..n-1, binomial(n-2*i-2,k-1)*sum(i=0..(n-k-1)/2, (-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1,n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2,n-2*i-2)*(n+j+(-2)*i-1)!,j,0,2*i))), n>1, a(1)=1.

a(n) ~ (-1)^(n+1) * n^(n-1) * s / (sqrt(1+sin(s)) * exp(n) * (1-sin(s))^n), where s = 0.73908513321516... (see A003957) is the root of the equation s = cos(s). - Vaclav Kotesovec, Jan 22 2014

Mathematica

a[n_] := Sum[ (-1)^k*Binomial[n+k-1, n-1] * Sum[ Binomial[n-2*i-2, k-1] * Sum[ (-1)^(j+i)*2^(-n+k-j+2*i+1)*StirlingS2[n-1, n+j-2*i-1] * Binomial[n+j-2*i-2, n-2*i-2]*(n+j-2*i-1)!, {j, 0, 2*i}], {i, 0, (n-k-1)/2}], {k, 1, n-1}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Feb 22 2013 *)
Rest[CoefficientList[InverseSeries[Series[x*(Sec[x]+Tan[x]), {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(Maxima) a(n):=if n=1 then 1 else
sum((-1)^k*binomial(n+k-1, n-1)*sum(binomial(n-2*i-2, k-1)*sum((-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1, n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2, n-2*i-2)*(n+j+(-2)*i-1)!, j, 0, 2*i), i, 0, (n-k-1)/2), k, 1, n-1)

Crossrefs

Cf. A003957.

Sequence in context: A180747 A227457 A134200 * A038037 A138212 A134261

Adjacent sequences: A217063 A217064 A217065 * A217067 A217068 A217069

Keyword

sign

Author

Vladimir Kruchinin, Sep 26 2012

Status

approved