%I #8 Dec 08 2020 11:54:42
%S 1,2,3,7,27,137,873,6667,59427,605297,6936273,88315027,1236909627,
%T 18898578857,312811478073,5575974948187,106493278764627,
%U 2169463248814817,46958188870266273,1076202413702266147,26035005959162168427,662975982648919697177,17726672005071080580873
%N P_n(1) + Q_n(1) (see A155100 and A104035), defining Q_{-1} = 0.
%F a(n) ~ (1 + sqrt(2)) * 2^(2*n + 2) * n^(n + 1/2) / (Pi^(n + 1/2) * exp(n)). - _Vaclav Kotesovec_, Dec 08 2020
%t p[n_, u_] := D[Tan[x], {x, n}] /. Tan[x] -> u /. Sec[x] -> Sqrt[1+u^2] // Expand; p[-1, u_] = 1; t[n_, k_] := t[n, k] = k*t[n-1, k-1]+(k+1)*t[n-1, k+1]; t[0, 0] = 1; t[0, _] = 0; t[-1, _] = 0; q[n_, u_] := Sum[t[n, k]*u^k, {k, 0, n}]; a[n_] := p[n, 1]+q[n, 1]; Table[a[n], {n, -1, 21}] (* _Jean-François Alcover_, Feb 05 2014 *)
%t nmax = 20; Join[{1}, CoefficientList[Series[(Sin[x] + Cos[x] + 1)/(Cos[x] - Sin[x]), {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Dec 08 2020 *)
%Y For P_n(1) - Q_n(1) see A007836.
%Y For P_n(0) + Q_n(0) see A000111.
%Y For P_n(1) * Q_n(1) see A156143.
%K nonn
%O -1,2
%A _N. J. A. Sloane_, Nov 06 2009