A136559
G.f.: A(x) = Sum_{n>=0} arctanh( 2^(2n+1)*x )^(2n+1) / (2n+1)!; a power series in x with integer coefficients.
Original entry on oeis.org
2, 0, 88, 0, 285088, 0, 112173964160, 0, 6667221644498203136, 0, 66605167708510907980664608768, 0, 120169056821375322042225614651624227643392, 0, 41233460218449924405779202537032142206549563511026450432, 0, 2796406262888046560966728498782777223041570797904775508376399120263413760
Offset: 1
G.f.: A(x) = 2*x + 88*x^3 + 285088*x^5 + 112173964160*x^7 + ...
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m:=30;
R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (&+[Argtanh(2^(2*j+1)*x)^(2*j+1)/Factorial(2*j+1): j in [0..m+2]]) )); // G. C. Greubel, Mar 15 2021
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Rest@With[{m=30}, CoefficientList[Series[Sum[ArcTanh[2^(2*j+1)*x]^(2*j+1)/(2*j + 1)!, {j,0,m+2}], {x,0,m}], x]] (* G. C. Greubel, Mar 15 2021 *)
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{a(n)=polcoeff(sum(k=0,n\2,atanh(2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}
A136647
G.f.: A(x) = Sum_{n>=0} arcsinh( 2^n*x )^n / n!; a power series in x with integer coefficients.
Original entry on oeis.org
1, 2, 8, 84, 2688, 276892, 94978048, 111457917800, 457117679616000, 6660816097416169260, 349290546231751288553472, 66597307693046550483175282456, 46556113319179632622352835689840640
Offset: 0
G.f.: A(x) = 1 + 2*x + 8*x^2 + 84*x^3 + 2688*x^4 + 276892*x^5 + ...
This is a special application of the following identity.
Let F(x), G(x), be power series in x such that F(0)=1, G(0)=1, then
Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! =
Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n).
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m:=30;
R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (&+[Argsinh(2^j*x)^j/Factorial(j): j in [0..m+2]]) )); // G. C. Greubel, Mar 15 2021
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m:=30;
S:=series( add( arcsinh(2^j*x)^j/j! , j=0..m+2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 15 2021
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With[{m=30}, CoefficientList[Series[Sum[ArcSinh[2^j*x]^j/j!, {j,0,m+2}], {x,0,m}], x]] (* G. C. Greubel, Mar 15 2021 *)
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{a(n)=polcoeff(sum(k=0,n, asinh(2^k*x +x*O(x^n))^k/k!),n)}
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{a(n)=polcoeff((sqrt(1+x^2)+x+x*O(x^n))^(2^n),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.
Original entry on oeis.org
2, 84, 2516412, 25131689308776, 73459034127708442263660, 59475400379433834763260101514326040, 12984879931670595437855043594849682375333268239320
Offset: 1
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 + ...
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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
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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
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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 *)
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{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)}
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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
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