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Search: id:A039683
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| A039683 |
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Double Pochhammer triangle: expansion of x(x+2)(x+4)..(x+2n-2). |
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+0
10
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| 1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
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REFERENCES
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Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
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LINKS
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W. Lang, First 9 rows and comment.
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FORMULA
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a(n, m) = a(n-1, m-1) - 2*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+2*x))/2)^m)/m!.
E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 11 2003
a(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
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EXAMPLE
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{1}, {2,1}, {8,6,1}, {48,44,12,1}, ...
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MATHEMATICA
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Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]
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CROSSREFS
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First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1). Cf. A051141, A051142.
Adjacent sequences: A039680 A039681 A039682 this_sequence A039684 A039685 A039686
Sequence in context: A011244 A008517 A114193 this_sequence A108084 A108085 A011135
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KEYWORD
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sign,tabl
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AUTHOR
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wouter.meeussen(AT)pandora.be
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EXTENSIONS
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Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).
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