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Search: id:A123242
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| A123242 |
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An even-odd switched polynomial recursion between a Bessel-like polynomial and a Poncelet-like polynomial to give a new triangular array: even:p(k, x)=2*x*p(k - 1, x) + (1 - x^2)*p(k - 2, x) odd:p(k, x)=2*(k - 1)*p(k - 1, x) - x*p(k - 2, x). |
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+0
1
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| 1, 1, 1, 1, 2, 1, 4, 7, 3, 1, 10, 14, 4, -1, 8, 76, 105, 29, -8, 1, 26, 165, 204, 43, -20, 1, 12, 304, 1904, 2343, 487, -232, 12, 1, 50, 772, 3986, 4564, 750, -506, 44, -1, 16, 788, 12048, 61872, 70681, 11513, -7864, 692, -16, 1, 82, 2347, 28032, 127536, 138126, 17956, -16434, 1889, -76, 1
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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The Bessel recursive polynomial in its two steps to advance in power, is very like spin pairs. The Poncelet recursion behaves as if it were two coupled states from a half plane to a disk. The total result is like a two-particle system emitting or absorbing a couple of plane waves: a radial one dimensional box quantum system like a Gopal phonon or a Ulam soliton.
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REFERENCES
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E. S. R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966, pages 36-40.
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 110
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FORMULA
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p(k, x) = If[Mod[k, 2] == 1, 2*(k - 1)*p(k - 1, x) - x*p(k - 2, x), 2*x*p(k - 1, x) + (1 - x^2)*p(k - 2, x)]
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EXAMPLE
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Row sum:
Table[Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[n, x], x]]}], {n, 0, 15}]
{1, 2, 4, 14, 28, 210, 420, 4830, 9660, 149730, 299460, 5839470, 11678940,
274455090, 548910180, 15095029950}
Triangle:
{1},
{1, 1},
{1, 2, 1},
{4, 7, 3},
{1, 10, 14, 4, -1},
{8, 76, 105, 29, -8},
{1, 26, 165, 204, 43, -20, 1}
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MATHEMATICA
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p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = If[Mod[k, 2] == 1, 2*(k - 1)*p[k - 1, x] - x*p[k - 2, x], 2*x*p[k - 1, x] + (1 - x^2)*p[k - 2, x]]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
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CROSSREFS
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Sequence in context: A072010 A123360 A072015 this_sequence A139769 A007839 A045625
Adjacent sequences: A123239 A123240 A123241 this_sequence A123243 A123244 A123245
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KEYWORD
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uned,tabf,sign
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 07 2006
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