%I A099376
%S A099376 0,1,4,14,48,165,572,2002,7072,25194,90440,326876,1188640,4345965,
%T A099376 15967980,58929450,218349120,811985790,3029594040,11338026180,
%U A099376 42550029600,160094486370,603784920024,2282138106804,8643460269248
%N A099376 An inverse Chebyshev transform of x^3.
%C A099376 The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial 
               transform of (-1)^n*A003518(n). Second binomial transform of expansion 
               of x^3c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed 
               to x^3 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)). 
               For a sequence b(n), this corresponds to taking sum{k=0..floor(n/
               2), C(n-k,k)(-1)^k*b(n-2k)}, or sum{k=0..n, C((n+k)/2,k)b(k)(-1)^((n-k)/
               2)(1+(-1)^(n-k))/2}.
%F A099376 G.f.: (1-2x)^4(sqrt((1+2x)/(1-2x))-1)^8/(256x^5); a(n)=sum{k=0..n, (k+1)C(n, 
               (n-k)/2)(-1)^k(C(3, k)-3C(2, k)+3C(1, k)-C(0, k))(1+(-1)^(n-k))/(n+k+2)}.
%F A099376 a(n)=A002057(n-1). - Michael Somos Jul 31 2005
%F A099376 Given an ellipse with eccentricity e and major and minor axis a and b 
               respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/
               2)^8 +48*(e/2)^10 +... - Michael Somos Apr 11 2007
%F A099376 E.g.f.: exp(2x)(Bessel_I(1,2x)-Bessel_I(3,2x)); - Paul Barry (pbarry(AT)wit.ie), 
               Jun 04 2007
%o A099376 (PARI) {a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* Michael Somos 
               Apr 11 2007 */
%Y A099376 Cf. A003518, A000245.
%Y A099376 Sequence in context: A094827 A094667 A002057 this_sequence A047048 A071745 
               A071749
%Y A099376 Adjacent sequences: A099373 A099374 A099375 this_sequence A099377 A099378 
               A099379
%K A099376 easy,nonn
%O A099376 0,3
%A A099376 Paul Barry (pbarry(AT)wit.ie), Oct 13 2004