Revision History for A123961
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#14 by Andrey Zabolotskiy at Wed Feb 24 08:14:55 EST 2021
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#13 by Andrey Zabolotskiy at Wed Feb 24 08:14:31 EST 2021
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| REFERENCES
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Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModularEquation.html">Modular Equation</a>
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| LINKS
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Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModularEquation.html">Modular Equation</a>
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approved
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#12 by N. J. A. Sloane at Sat Feb 20 00:36:39 EST 2021
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#11 by G. C. Greubel at Fri Feb 19 22:54:35 EST 2021
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#10 by G. C. Greubel at Fri Feb 19 22:51:40 EST 2021
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| REFERENCES
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Eric Weisstein's World of Mathematics, "Modular, <a Equation." href="http://mathworld.wolfram.com/ModularEquation.html">Modular Equation</a>
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| STATUS
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proposed
editing
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#9 by G. C. Greubel at Fri Feb 19 22:46:11 EST 2021
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#8 by G. C. Greubel at Fri Feb 19 22:45:34 EST 2021
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| NAME
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A triangular sequence from the omega2 Jacobian Elliptic Modular function.
Triangle T(n, k) = k^2*(1+n)^2 - 4*n, read by rows.
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| DATA
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0, -4, 0, -8, 1, 28, -12, 4, 52, 132, -16, 9, 84, 209, 384, -20, 16, 124, 304, 556, 880, -24, 25, 172, 417, 760, 1201, 1740, -28, 36, 228, 548, 996, 1572, 2276, 3108, -32, 49, 292, 697, 1264, 1993, 2884, 3937, 5152, -36, 64, 364, 864, 1564, 2464, 3564, 4864, 6364, 8064, -40, 81, 444, 1049, 1896, 2985, 4316, 5889, 7704, 9761, 12060
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| OFFSET
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10,2
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| COMMENTS
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Normally these functions are taken as implicit polynomials in two variables set equal to zero. Row sum: Table[Sum[t[n, m], {n, 0, m}], {m, 0, 10}] {0, -4, 21, 176, 670, 1860, 4291, 8736, 16236, 28140, 46145}
A triangular sequence formed from the omega2 Jacobian Elliptic Modular function.
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| LINKS
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G. C. Greubel, <a href="/A123961/b123961.txt">Table of n, a(n) for n = 0..5150</a>
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| FORMULA
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tT(n,m) =n, k) = k^2*(1 + m+n)^2 - 4*mn.
Sum_{k=0..n} T(n, k) = (n*(n+1)/6)*( 2*n^3 + 5*n^2 + 4*n - 23 ). (n+1)^2 * A000330(n) - 8 * A000217(n). - G. C. Greubel, Feb 19 2021
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| EXAMPLE
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{0},
0;
{- -4, , 0},;
{- -8, , 1, , 28},;
{- -12, , 4, , 52, , 132},;
{- -16, , 9,, 84, , 209, , 384},;
{- -20, 16, 124, , 304, , 556, , 880},;
{- -24, 25, 172, , 417, , 760, 1201, 1740},;
{- -28, 36, 228, , 548, , 996, 1572, 2276, 3108},;
{- -32, 49, 292, , 697,, 1264, 1993, 2884, 3937, 5152},;
{- -36, 64, 364, , 864, 1564, 2464, 3564, 4864, 6364, 8064},;
{- -40, 81, 444, 1049, 1896, 2985, 4316, 5889, 7704, 9761, 12060};
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| MATHEMATICA
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t[n_, m_] = n^2*(1 + m)^2 - 4*m a = Table[Table[t[n, m], {n, 0, m}], {m, 0, 10}] Flatten[a]
T[n_, k_]:= k^2*(1+n)^2 - 4*n;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
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| PROG
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(Sage) flatten([[k^2*(n+1)^2 - 4*n for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021
(Magma) [k^2*(n+1)^2 - 4*n: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021
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| CROSSREFS
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Cf. A000217, A000330.
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| KEYWORD
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uned,tabl,sign
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| EXTENSIONS
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Edited by G. C. Greubel, Feb 19 2021
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approved
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#7 by Charles R Greathouse IV at Wed Mar 12 16:37:02 EDT 2014
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| AUTHOR
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_Roger L. Bagula_, Oct 28 2006
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Discussion
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| Wed Mar 12
| 16:37
| OEIS Server: https://oeis.org/edit/global/2126
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#6 by Russ Cox at Fri Mar 30 18:49:18 EDT 2012
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| AUTHOR
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_Roger Bagula (rlbagulatftn(AT)yahoo.com), _, Oct 28 2006
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Discussion
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| Fri Mar 30
| 18:49
| OEIS Server: https://oeis.org/edit/global/236
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#5 by N. J. A. Sloane at Sat Jul 31 03:00:00 EDT 2010
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| KEYWORD
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uned,probation,tabl,sign,new
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