A217582 E.g.f. 1/2*sqrt(sec(2*x))-1/2, (even part).

0, 1, 14, 556, 43784, 5723536, 1119636704, 306179943616, 111530881745024, 52199950088663296, 30524582707646303744, 21808622670887632792576, 18692756653071421750052864, 18931292094375391032677011456, 22364730782577535845815428112384

Offset

0,3

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = sum(m=1..2*n, ((-1)^m*C(2*m-2,m-1)*sum(k=m..n, C(k-1,m-1)* sum(j=2*k..2*n, C(j-1,2*k-1)*j!*2^(2*n-m-j)*(-1)^(n+k+j+1)* stirling2(2*n,j))))/m).

G.f.: T(0)/2 -1/2, where T(k) = 1 - x*(2*k+1)*(2*k+2)/( x*(2*k+1)*(2*k+2) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013

a(n) ~ 2^(6*n+1)*n^(2*n)/(Pi^(2*n+1/2)*exp(2*n)). - Vaclav Kotesovec, Nov 07 2013

Mathematica

a[n_] := Sum[ ((-1)^m*Binomial[2*m-2, m-1]* Sum[ Binomial[k-1, m-1]* Sum[ Binomial[j-1, 2*k-1]* j!*2^(2*n-m-j)*(-1)^(n+k+j+1)*StirlingS2[2*n, j], {j, 2*k, 2*n}], {k, m, n}])/m, {m, 1, 2*n}]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Feb 22 2013, translated from Maxima *)
With[{nn=30}, Take[CoefficientList[Series[(Sqrt[Sec[2x]]-1)/2, {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Mar 28 2024 *)

Prog

(Maxima) a(n):=sum(((-1)^m*binomial(2*m-2, m-1)*sum(binomial(k-1, m-1)* sum(binomial(j-1, 2*k-1)*j!*2^(2*n-m-j)*(-1)^(n+k+j+1) *stirling2(2*n, j), j, 2*k, 2*n), k, m, n))/m, m, 1, 2*n);

Crossrefs

Sequence in context: A109773 A294835 A205582 * A002010 A166185 A361695

Adjacent sequences: A217579 A217580 A217581 * A217583 A217584 A217585

Keyword

nonn

Author

Vladimir Kruchinin, Oct 07 2012

Status

approved