A185142 E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! is inverse function to x*cos(x).

1, 3, 85, 6727, 1045161, 268614731, 103164046973, 55349799523215, 39541660762919761, 36286594559417097619, 41598050801794414418085, 58257277349451323696625623, 97872074004750264647795154425

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..200

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Formula

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.

a(n) = [x^(2*n)/(2*n)!] 1/cos(x)^(2*n+1). - Paul D. Hanna, Jan 23 2012

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

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A196873.

Sequence in context: A156879 A120264 A292830 * A279020 A302947 A326948

Adjacent sequences: A185139 A185140 A185141 * A185143 A185144 A185145

Keyword

nonn

Author

Vladimir Kruchinin, Jan 23 2012

Status

approved