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Search: id:A157285
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| A157285 |
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A general system of triangular Hahn like weights:m=2; 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, 2, 2, 6, 12, 18, 28, 84, 336, 1456, 210, 840, 6300, 88200, 1874250, 2604, 13020, 156240, 4843440, 377788320, 59010535584, 54684, 328104, 5741820, 329197680, 63946649340, 39774815889480, 61856467844122980, 1984248, 13889736
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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The row sums are:
{1, 4, 36, 1904, 1969800, 59393339208, 61896306941984088,
2728499296658565047099328, 6119714812198965904564001339406480,
818784237539437193511087327034940829681115800,
7501705327173906348158617454088093174955631623738333820952,...}.
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=2;
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},
{2, 2},
{6, 12, 18},
{28, 84, 336, 1456},
{210, 840, 6300, 88200, 1874250},
{2604, 13020, 156240, 4843440, 377788320, 59010535584},
{54684, 328104, 5741820, 329197680, 63946649340, 39774815889480, 61856467844122980},
{1984248, 13889736, 333353664, 31668598080, 12667439232000, 21288898373299200, 139144239767883571200, 2728360131117199333171200},
{126495810, 1011966480, 31876944120, 4654033841520, 3403262246611500, 12744536461110745200, 238635072966068148497400, 20426684975749501375080445200, 6119694385275342334152147703859250},
{14364301980, 129278717820, 5171148712800, 1098007243351200, 1350548909321976000, 9968401499705504856000, 441467274416957791722720000, 113117348924117559383594712480000, 152166588946212182058405443175220800000, 818784085372735129508503906900004331163200000},
{2938936185108, 29389361851080, 1454773411628460, 430612929842024160, 837219188845355473080, 11162810888712893593670256, 1033592567213148600071913178680, 656249183399296430956859156500750560, 2734371570153352487349561862217329145802060, 67515347343678541165338503083154747742832860597480, 7501705259658556270107849885180099714946376202577412660028}
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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: A079005 A156992 A054481 this_sequence A035615 A115962 A019311
Adjacent sequences: A157282 A157283 A157284 this_sequence A157286 A157287 A157288
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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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