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%I A157315 #9 Mar 16 2021 01:26:22
%S A157315 2,84,2516412,25131689308776,73459034127708442263660,
%T A157315 59475400379433834763260101514326040,
%U A157315 12984879931670595437855043594849682375333268239320
%N A157315 G.f.: A(x) = sin( Sum_{n>=0} 2^((2n+1)^2) * C(2n,n)/4^n * x^(2n+1)/(2n+1) ); alternating zeros omitted.
%C A157315 Compare g.f. to the expansion of the inverse sine of x:
%C A157315 arcsin(x) = Sum_{n>=0} C(2n,n)/4^n * x^(2n+1)/(2n+1).
%H A157315 G. C. Greubel, <a href="/A157315/b157315.txt">Table of n, a(n) for n = 1..29</a>
%e A157315 G.f.: A(x) = 2*x + 84*x^3 + 2516412*x^5 + 25131689308776*x^7 + ...
%e A157315 The inverse sine of A(x) begins:
%e A157315 arcsin(A(x)) = 2*x + 2^9*(2/4)*x^3/3 + 2^25*(6/4^2)*x^5/5 + 2^49*(20/4^3)*x^7/7 + 2^81*(70/4^4)*x^9/9 + ...
%p A157315 m := 30;
%p A157315 S := series( sin(add(2^(4*j^2+2*j+1)*binomial(2*j,j)*x^(2*j+1)/(2*j+1), j = 0..m+2)), x, m+1);
%p A157315 seq(coeff(S, x, 2*j+1), j = 0..m/2); # _G. C. Greubel_, Mar 16 2021
%t A157315 With[{m = 30}, CoefficientList[Series[Sin[Sum[2^(4*n^2+2*n+1)*((n+1)/(2*n+1)) *CatalanNumber[n]*x^(2*n+1), {n,0,m+2}]], {x,0,m}], x]][[2 ;; ;; 2 ]] (* _G. C. Greubel_, Mar 16 2021 *)
%o A157315 (PARI) {a(n)=polcoeff(sin(sum(m=0,n\2,2^((2*m+1)^2)*binomial(2*m,m)/4^m*x^(2*m+1)/(2*m+1))+x*O(x^n)),n)}
%o A157315 (Magma)
%o A157315 m:=30;
%o A157315 R<x>:=PowerSeriesRing(Rationals(), m);
%o A157315 b:=Coefficients(R!( Sin( (&+[2^(4*j^2+2*j+1)*Binomial(2*j,j)*x^(2*j+1)/(2*j+1): j in [0..m+2]]) ) ));
%o A157315 [b[2*n-1]: n in [1..Floor((m-2)/2)]]; // _G. C. Greubel_, Mar 16 2021
%o A157315 (Sage)
%o A157315 m=30
%o A157315 def A157315_list(prec):
%o A157315     P.<x> = PowerSeriesRing(QQ, prec)
%o A157315     return P( sin( sum(2^(4*j^2+2*j+1)*binomial(2*j,j)*x^(2*j+1)/(2*j+1) for j in [0..m+2])) ).list()
%o A157315 a=A157315_list(m); [a[2*n+1] for n in (0..(m-2)/2)] # _G. C. Greubel_, Mar 16 2021
%Y A157315 Cf. A000984 (C(2n, n)), A136558, A155200.
%K A157315 nonn
%O A157315 1,1
%A A157315 _Paul D. Hanna_, Mar 17 2009

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