Revision History for A214377
(Underlined text is an addition;
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Showing entries 1-10
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#42 by N. J. A. Sloane at Sat Mar 23 20:23:52 EDT 2024
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#41 by Jon E. Schoenfield at Sat Mar 23 17:53:12 EDT 2024
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#40 by Jon E. Schoenfield at Sat Mar 23 17:53:10 EDT 2024
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| FORMULA
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G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6],[5/2],108*z^2); a(n)= ) = Integral_{x=0.....sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3);).
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| EXAMPLE
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G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10296*x^5 + 86016*x^6 +... + ... where A(x) = 1 + 4*x*A(x)^(3/2).
A(x)^(3/2) = 1 + 6*x + 42*x^2 + 320*x^3 + 2574*x^4 + 21504*x^5 + 184756*x^6 +... + ...
A(x)^(1/2) = 1 + 2*x + 10*x^2 + 64*x^3 + 462*x^4 + 3584*x^5 + 29172*x^6 +...+ + ... + A078531(n)*x^n +... + ...
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| STATUS
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proposed
editing
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#39 by Karol A. Penson at Sat Mar 23 17:38:17 EDT 2024
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#38 by Karol A. Penson at Sat Mar 23 17:36:42 EDT 2024
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| FORMULA
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G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6],[5/2],108*z^2); ); a(n)= Integral_{x=0...sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
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#37 by Karol A. Penson at Sat Mar 23 17:34:40 EDT 2024
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| FORMULA
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G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6], [],[5/2],108*z^2); ); a(n) = )= Integral_{x=0...sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
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#36 by Karol A. Penson at Sat Mar 23 17:32:13 EDT 2024
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| FORMULA
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G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6], [5/2], ],108*z^2); ); a(n) = Integral_{x=0...sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
where g1(x) = (18 - sqrt(324 - 3*x^2))^(2/3) and g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3);
g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3);
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#35 by Karol A. Penson at Sat Mar 23 17:27:27 EDT 2024
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| FORMULA
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From Karol A. Penson, Mar 23 2024: (Start)
G.f. = 1/(48*z^2) - 2F1([-2/3, -1/3], [-1/2], 108*z^2)/(48*z^2) + 4*z*2F1([5/6, 7/6], [5/2], 108*z^2); a(n) = Integral_{x=0...sqrt(108)} x^n*W(x), with W(x) = ((72*(g1(x) - g2(x)) + x^2*(-g1(x) + g2(x)) + 4*sqrt(-3*x^2 + 324)*(g1(x) + g2(x)))*3^(1/6))/(96*Pi*(x^2)^(5/6)),
where g1(x) = (18 - sqrt(324 - 3*x^2))^(2/3) and g2(x) = (18 + sqrt(324 - 3*x^2))^(2/3);
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, sqrt(108)). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = sqrt(108). For x -> sqrt(108), W'(x) tends to -infinity. (End)
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| STATUS
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approved
editing
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#34 by Michael De Vlieger at Fri Feb 03 12:19:56 EST 2023
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#33 by Joerg Arndt at Fri Feb 03 11:24:48 EST 2023
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