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.

2, 84, 2516412, 25131689308776, 73459034127708442263660, 59475400379433834763260101514326040, 12984879931670595437855043594849682375333268239320

Offset: 1

Author

Paul D. Hanna, Mar 17 2009

Keywords

• nonn

Comments

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

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

Examples

			G.f.: A(x) = 2*x + 84*x^3 + 2516412*x^5 + 25131689308776*x^7 + ...
The inverse sine of A(x) begins:
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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..29

Crossrefs

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

Programs

Magma

m:=30;
R:=PowerSeriesRing(Rationals(), m);
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]]) ) ));
[b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Mar 16 2021

Maple

m := 30;
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);
seq(coeff(S, x, 2*j+1), j = 0..m/2); # G. C. Greubel, Mar 16 2021

Mathematica

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 *)

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)}

Sage

m=30
def A157315_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    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()
a=A157315_list(m); [a[2*n+1] for n in (0..(m-2)/2)] # G. C. Greubel, Mar 16 2021