A192063 E.g.f. 1-sqrt(cos(2*x)) (even part).

0, 2, 4, 152, 8944, 933152, 151557184, 35402298752, 11250504212224, 4668840721981952, 2451963626804184064, 1589715293557268682752, 1247113599659216858312704, 1164315843409068590677041152, 1275742921918699248939411718144, 1621172561651122048792832473137152

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = sum((k=1..2*n, binomial(2*k-2,k-1)*2^(2*n-2*k+2)*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))/k).

a(n) = 2*sum(k=1..2*n, C(k-1)*sum(i=0..k-1, (i-k)^(2*n)*binomial(2*k,i)*(-1)^(n+k-i))*2^(2*n-3*k+1)), where C(k) = A000108(k). - Vladimir Kruchinin, Oct 05 2012

G.f.: 1 - 1/U(0) where U(k)= 1 - (2*k-1)*(2*k+2)*x/U(k+1); (continued fraction, due to T. J. Stieltjes). - Sergei N. Gladkovskii, Nov 09 2012

G.f.: T(0)+1, 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

a(n) ~ (2*n)! * 2^(4*n-2)/(n^(3/2)*Pi^(2*n)). - Vaclav Kotesovec, Nov 07 2013

Mathematica

Table[n!*SeriesCoefficient[1-Sqrt[Cos[2*x]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Nov 07 2013 *)
With[{nn=40}, Take[CoefficientList[Series[1-Sqrt[Cos[2x]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Nov 01 2021 *)

Prog

(Maxima)
a(n):=sum((binomial(2*k-2, k-1)*2^(2*n-2*k+2)*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, k, 1, 2*n);

Crossrefs

Sequence in context: A018517 A018539 A018544 * A326796 A018558 A296463

Adjacent sequences: A192060 A192061 A192062 * A192064 A192065 A192066

Keyword

nonn

Author

Vladimir Kruchinin, Jun 22 2011

Extensions

Corrected and extended by Harvey P. Dale, Nov 01 2021

Status

approved