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Search: id:A132062
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| A132062 |
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Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497. |
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1
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| 1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 0, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 0
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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This is a Jabotinsky type exponential convolution triangle related to A001147 (double factorials). For Jabotinsky type triangles See the D. E. Knuth reference given under A039692.
The subtriangle (n>=m>=1) is A001497(n,m) (Bessel).
For the combinatorial interpretation in terms of unordered forests of increasing plane trees see the W. Lang comment and example under A001497.
This is a special type of Sheffer triangle. See the S. Roman reference given under A048854 (the notation here differs).
This triangle (or the A001497 subtriangle) appears as generalized Stirling numbers of the second kind, S2p(-1,n,m):=S2(-k;m,m)*(-1)^(n-m) for k=1, eqs. (27)-(29) of the W. Lang reference.
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LINKS
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W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
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FORMULA
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a(n,m)=0 if n<m; a(n,0)=1 if n=1, else 0; a(n,m)=(2*(n-1)-m)*a(n-1,m) + a(n-1,m-1).
E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).
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EXAMPLE
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[1];[0,1];[0,1,1];[0,3,3,1];[0,15,15,6,1];...
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CROSSREFS
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Columns m=1: A001147.
Row sums give [1, A001515]. Alternating row sums give [1, -A000806].
Sequence in context: A135009 A092747 A122850 this_sequence A065547 A143333 A065551
Adjacent sequences: A132059 A132060 A132061 this_sequence A132063 A132064 A132065
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Sep 14 2007
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