%I #9 Mar 16 2021 01:26:22

%S 2,84,2516412,25131689308776,73459034127708442263660,

%T 59475400379433834763260101514326040,

%U 12984879931670595437855043594849682375333268239320

%N 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 Compare g.f. to the expansion of the inverse sine of x:

%C arcsin(x) = Sum_{n>=0} C(2n,n)/4^n * x^(2n+1)/(2n+1).

%H G. C. Greubel, <a href="/A157315/b157315.txt">Table of n, a(n) for n = 1..29</a>

%e G.f.: A(x) = 2*x + 84*x^3 + 2516412*x^5 + 25131689308776*x^7 + ...

%e The inverse sine of A(x) begins:

%e 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 m := 30;

%p 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 seq(coeff(S, x, 2*j+1), j = 0..m/2); # _G. C. Greubel_, Mar 16 2021

%t 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 (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 (Magma)

%o m:=30;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o 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 [b[2*n-1]: n in [1..Floor((m-2)/2)]]; // _G. C. Greubel_, Mar 16 2021

%o (Sage)

%o m=30

%o def A157315_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o 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 a=A157315_list(m); [a[2*n+1] for n in (0..(m-2)/2)] # _G. C. Greubel_, Mar 16 2021

%Y Cf. A000984 (C(2n, n)), A136558, A155200.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Mar 17 2009