A190123 - OEIS

%I #16 Sep 08 2022 08:45:56

%S 1,1,2,7,32,177,1184,9175,81280,810081,8967168,109200551,1450641408,

%T 20876239633,323542851584,5372445971063,95157141241856,

%U 1790769169786049,35682993123753984,750523142329023815,16616642326426025984,386288476226459349361,9407703499451286945792

%N Expansion of e.g.f.: 1/(1-tan(sin(x))).

%H G. C. Greubel, <a href="/A190123/b190123.txt">Table of n, a(n) for n = 0..435</a>

%F a(n) = Sum_{m=1..n} Sum_{k=m..n} (((-1)^(k-m)+1)*(Sum_{j=m..k} binomial(j-1,m-1)*j!*2^(k-j-1)*stirling2(k,j)*(-1)^((m+k)/2+j),j,m,k))*((-1)^(n-k)+1)*Sum_{i=0..k/2} (2*i-k)^n*binomial(k,i)*(-1)^((n+k)/2-i)))/(2^k*k!))), n>0, a(0)=1.

%p a:=series(1/(1-tan(sin(x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 27 2019

%t With[{nmax = 50}, CoefficientList[Series[1/(1 - Tan[Sin[x]]), {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Dec 29 2017 *)

%o (Maxima)

%o a(n):=sum(sum((((-1)^(k-m)+1)*(sum(binomial(j-1,m-1)*j!*2^(k-j-1)*stirling2(k,j)*(-1)^((m+k)/2+j),j,m,k))*((-1)^(n-k)+1)*sum((2*i-k)^n*binomial(k,i)*(-1)^((n+k)/2-i),i,0,k/2))/(2^k*k!),k,m,n),m,1,n);

%o (PARI) x='x+O('x^30); Vec(serlaplace(1/(1-tan(sin(x))))) \\ _G. C. Greubel_, Dec 29 2017

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(  1/(1 - Tan(Sin(x)))  )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Nov 07 2018

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, May 04 2011