A185142 - OEIS

%I #33 Oct 18 2024 20:29:27

%S 1,3,85,6727,1045161,268614731,103164046973,55349799523215,

%T 39541660762919761,36286594559417097619,41598050801794414418085,

%U 58257277349451323696625623,97872074004750264647795154425,194268677575370080513687519965147,449782936650769586164505607701592781

%N E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! is inverse function to x*cos(x).

%H G. C. Greubel, <a href="/A185142/b185142.txt">Table of n, a(n) for n = 0..200</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

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

%F a(n) = [x^(2*n)/(2*n)!] 1/cos(x)^(2*n+1). - _Paul D. Hanna_, Jan 23 2012

%F a(n) = (2*n+1) * A196873(n) for n>=1, where e.g.f. G(x) of A196873 satisfies: G(x*cos(x)) = 1/cos(x). - _Paul D. Hanna_, Jan 23 2012

%F a(n) = Sum_{k=1..2*n} (binomial(2*n+k,2*n)*Sum_{i=0..k-1} (i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+k-i)))/2^(k-1), with n>0, a(0)=1. - _Vladimir Kruchinin_, Oct 08 2012

%t a[n_] := Sum[ (Binomial[2*n + k, 2*n]*Sum[ (i - k)^(2*n)*Binomial[2*k, i]*(-1)^(n + k - i), {i, 0, k - 1}])/2^(k - 1), {k, 1, 2*n}]; a[0] = 1; Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Feb 21 2013, translated from Maxima *)

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

%o (PARI) {a(n)=if(n==0,1,2*sum(k=1,2*n,binomial(2*n+k,2*n)*sum(j=1,k,sum(i=0,floor((j-1)/2),(j-2*i)^(2*n)*binomial(j,i))*binomial(k,j)*(-1)^(n-j)/2^j)))}

%o (PARI) {a(n)=(2*n)!*polcoeff(1/cos(x+x*O(x^(2*n+1)))^(2*n+1),2*n)}

%o (Maxima) a(n):=if n=0 then 1 else (sum((binomial(2*n+k,2*n)*sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+k-i),i,0,k-1))/2^(k-1),k,1,2*n));

%Y Cf. A196873.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Jan 23 2012