Revision History for A035002
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#92 by N. J. A. Sloane at Sat Oct 21 11:28:53 EDT 2023
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#91 by N. J. A. Sloane at Sat Oct 21 11:26:00 EDT 2023
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T(n,m) = 2^(-5 + m + n)*((m^3 + (-2*n - 12)*m^2 + (9*n^2 + 87*n + 83)*m + 9*n^2 + 185*n + 576)*hypergeom([2 - m, 1 - n], [-m - n], 3/4) - (-2 + m)*hypergeom([1 - n, 3 - m], [-m - n], 3/4)*(m^2 + (-5*n - 7)*m - 13*n - 48))*(m + n)!/(9*(2 + n)!*(1 + m)!) for n>=1 and m>=1. Mariusz Iwaniuk, Oct 21 2023
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proposed
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Discussion
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| Sat Oct 21
| 11:28
| N. J. A. Sloane: Calling it an "empirical observation" is a falsehood, no? Rejected.
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#90 by Mariusz Iwaniuk at Sat Oct 21 11:03:45 EDT 2023
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#89 by Mariusz Iwaniuk at Sat Oct 21 11:00:58 EDT 2023
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T(n,m) = T(n,m) = 2^(-5 + m + n)*((m^3 + (-2*n - 12)*m^2 + (9*n^2 + 87*n + 83)*m + 9*n^2 + 185*n + 576)*hypergeom([2 - m, 1 - n], [-m - n], 3/4) - (-2 + m)*hypergeom([1 - n, 3 - m], [-m - n], 3/4)*(m^2 + (-5*n - 7)*m - 13*n - 48))*(m + n)!/(9*(2 + n)!*(1 + m)!) for n>=1 and m>=1. - _. _Mariusz Iwaniuk_, Oct 21 2023
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#88 by Mariusz Iwaniuk at Sat Oct 21 10:59:58 EDT 2023
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T(n,m) = T(n,m) = 2^(-5 + m + n)*((m^3 + (-2*n - 12)*m^2 + (9*n^2 + 87*n + 83)*m + 9*n^2 + 185*n + 576)*hypergeom([2 - m, 1 - n], [-m - n], 3/4) - (-2 + m)*hypergeom([1 - n, 3 - m], [-m - n], 3/4)*(m^2 + (-5*n - 7)*m - 13*n - 48))*(m + n)!/(9*(2 + n)!*(1 + m)!) for n>=1 and m>=1. _. - _Mariusz Iwaniuk_, Oct 21 2023
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#87 by Mariusz Iwaniuk at Sat Oct 21 10:55:05 EDT 2023
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#86 by Mariusz Iwaniuk at Sat Oct 21 08:47:47 EDT 2023
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T(n,m) = 2^(-5 + m + n)*((m^3 + (-2*n - 12)*m^2 + (9*n^2 + 87*n + 83)*m + 9*n^2 + 185*n + 576)*hypergeom([2 - m, 1 - n], [-m - n], 3/4) - (-2 + m)*hypergeom([1 - n, 3 - m], [-m - n], 3/4)*(m^2 + (-5*n - 7)*m - 13*n - 48))*(m + n)!/(9*(2 + n)!*(1 + m)!) for n>=1 and m>=1. Mariusz Iwaniuk, Oct 21 2023
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approved
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Discussion
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| Sat Oct 21
| 08:49
| Mariusz Iwaniuk: I can't prove is only an empirical observation.
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| 10:11
| Joerg Arndt: "empirical observation": really? Didn't you just throw something at Wolfram Alpha and copy the output?
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| 10:18
| Joerg Arndt: Oh look: https://community.wolfram.com/groups/-/m/t/3053282
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| 10:27
| Mariusz Iwaniuk: To nasty Joerg Arndt .Yes Wolfram Alpha gave me the answer!
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#85 by Joerg Arndt at Thu Oct 19 05:05:28 EDT 2023
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T(n,m) = 2^(-5 - m - n)*csc((m + n)*Pi)*(4^(m + n)*GAMMA(-1 - m)*GAMMA(-2 - n)*((576 + m*(83 + (-12 + m)*m) + 185*n + (87 - 2*m)*m*n + 9*(1 + m)*n^2)*hypergeom([2 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n) - (-2 + m)*(-48 + m*(-7 + m - 5*n) - 13*n)*hypergeom([3 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n)) + 3^(1 + m + n)*GAMMA(-1 + m)*GAMMA(-2 + n)*(-(192 + m^3 - (-49 + n)*n + 6*m^2*(2 + n) + m*(83 + n*(41 + n)))*hypergeom([2 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n) + (2 + m)*(16 + 5*n + m*(7 + m + 3*n))*hypergeom([3 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n)))*sin(m*Pi)*sin(n*Pi)/(9*Pi) for m >= 1,n >= 1. - Mariusz Iwaniuk, Oct 12 2023
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#84 by Alexander R. Povolotsky at Thu Oct 19 00:06:28 EDT 2023
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T(n,m) = 2^(-5 - m - n)*csc((m + n)*Pi)*(4^(m + n)*GAMMA(-1 - m)*GAMMA(-2 - n)*((576 + m*(83 + (-12 + m)*m) + 185*n + (87 - 2*m)*m*n + 9*(1 + m)*n^2)*hypergeom([2 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n) - (-2 + m)*(-48 + m*(-7 + m - 5*n) - 13*n)*hypergeom([3 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n)) + 3^(1 + m + n)*GAMMA(-1 + m)*GAMMA(-2 + n)*(-(192 + m^3 - (-49 + n)*n + 6*m^2*(2 + n) + m*(83 + n*(41 + n)))*hypergeom([2 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n) + (2 + m)*(16 + 5*n + m*(7 + m + 3*n))*hypergeom([3 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n)))*sin(m*Pi)*sin(n*Pi)/(9*Pi) for m >= 1,n >= 1 - . - Mariusz Iwaniuk, Oct 12 2023
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Discussion
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| Thu Oct 19
| 02:31
| Joerg Arndt: I am skeptical about the machine-generated (and IMO overly complicated) formula. There are several formulas here which are all human-readable!
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#83 by Mariusz Iwaniuk at Wed Oct 11 18:01:02 EDT 2023
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T(n,m) = 2^(-5 - m - n)*csc((m + n)*Pi)*(4^(m + n)*GAMMA(-1 - m)*GAMMA(-2 - n)*((576 + m*(83 + (-12 + m)*m) + 185*n + (87 - 2*m)*m*n + 9*(1 + m)*n^2)*hypergeom([2 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n) - (-2 + m)*(-48 + m*(-7 + m - 5*n) - 13*n)*hypergeom([3 - m, 1 - n], [-m - n], 3/4)/GAMMA(-m - n)) + 3^(1 + m + n)*GAMMA(-1 + m)*GAMMA(-2 + n)*(-(192 + m^3 - (-49 + n)*n + 6*m^2*(2 + n) + m*(83 + n*(41 + n)))*hypergeom([2 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n) + (2 + m)*(16 + 5*n + m*(7 + m + 3*n))*hypergeom([3 + m, 1 + n], [m + n], 3/4)/GAMMA(m + n)))*sin(m*Pi)*sin(n*Pi)/(9*Pi) for m >= 1,n >= 1 - Mariusz Iwaniuk, Oct 12 2023
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approved
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Discussion
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| Wed Oct 11
| 18:03
| Mariusz Iwaniuk: See: https://community.wolfram.com/groups/-/m/t/3030771
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| Wed Oct 18
| 21:39
| OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review. If it is ready for review, please
visit https://oeis.org/draft/A035002 and click the button that reads
"These changes are ready for review by an OEIS Editor."
Thanks.
- The OEIS Server
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