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Search: id:A157284
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| A157284 |
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A general system of triangular Hahn like weights:m=1; q-factorial:t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; Hahn type weight factor: b(n,k,m)=If[n == 0, 1, ((m + 1)^n*t[n, m]*t[k, n - m])/(k!*(n - k)!)]. |
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| 1, 1, 1, 1, 2, 4, 1, 3, 15, 105, 1, 4, 36, 744, 29016, 1, 5, 70, 3010, 389795, 121226245, 1, 6, 120, 9120, 2736000, 3065414400, 10017774259200, 1, 7, 189, 22995, 13452075, 37781497845, 471626437599135, 20185139902805378865, 1, 8, 280, 50960
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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The row sums are:
{1, 2, 7, 124, 29801, 121619126, 10020842418847, 20185611567037951112,
1177044588221415514632464449, 2297271820821880031983085448483904042,
170310659716803009410138634830607646103800140551,...}.
This type of system is a neo-combinatorial system that mixes traditional factorial with q-form Gaussian factorials.
The Hahn application is hexagonal orthogonal polynomials and tiling systems.
I feel pretty lucky that I got a system of these weights that worked in Mathematica.
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REFERENCES
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J Baik, T Kriecherbauer, KDTR McLaughlin, P. D. Miller ,Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results, International Mathematics Research Notices (2003) 2003:821-858, doi:10.1155/S1073792803212125;http://intl-imrn.oxfordjournals.org/cgi/content/abstract/2003/15/821
V Gorin,Non-intersecting paths and Hahn orthogonal polynomial ensemble, Arxiv preprint arXiv:0708.2349, 2007
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FORMULA
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m=1;
q-factorial:t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
Hahn type weight factor:
b(n,k,m)=If[n == 0, 1, ((m + 1)^n*t[n, m]*t[k, n - m])/(k!*(n - k)!)].
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EXAMPLE
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{1},
{1, 1},
{1, 2, 4},
{1, 3, 15, 105},
{1, 4, 36, 744, 29016},
{1, 5, 70, 3010, 389795, 121226245},
{1, 6, 120, 9120, 2736000, 3065414400, 10017774259200},
{1, 7, 189, 22995, 13452075, 37781497845, 471626437599135, 20185139902805378865},
{1, 8, 280, 50960, 52234000, 308431323200, 10244546400088000, 1749976343076289328000, 1177042838234827583459440000},
{1, 9, 396, 102564, 170922906, 1899124408566, 140675741440117884, 66988441605973209518196, 186079002600246648495733068939, 2297271634742810443154153338805764581},
{1, 10, 540, 191520, 490674240, 9482770362240, 1399941388577491200, 1558908333099750239308800, 12531219161472352109173737830400, 656621317215054881215396509463495372800, 170310659060181679663863033233125976844488908800}
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MATHEMATICA
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Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, ((m + 1)^n*t[n, m]*t[k, n - m])/(k!*(n - k)!)];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
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CROSSREFS
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Sequence in context: A112973 A162303 A128570 this_sequence A143973 A011167 A014176
Adjacent sequences: A157281 A157282 A157283 this_sequence A157285 A157286 A157287
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KEYWORD
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nonn,tabl,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 26 2009
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