A217582 - OEIS

%I #21 Mar 28 2024 16:28:26

%S 0,1,14,556,43784,5723536,1119636704,306179943616,111530881745024,

%T 52199950088663296,30524582707646303744,21808622670887632792576,

%U 18692756653071421750052864,18931292094375391032677011456,22364730782577535845815428112384

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

%F 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).

%F 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

%F a(n) ~ 2^(6*n+1)*n^(2*n)/(Pi^(2*n+1/2)*exp(2*n)). - _Vaclav Kotesovec_, Nov 07 2013

%t 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 *)

%t 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 *)

%o (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);

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Oct 07 2012