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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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8
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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
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OFFSET
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0,8
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COMMENTS
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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.
Also the Bell transform of the double factorial of odd numbers A001147. For the Bell transform of the double factorial of even numbers A000165 see A039683. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015
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REFERENCES
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Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]
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LINKS
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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).
a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).
The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)
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EXAMPLE
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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]
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MAPLE
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# The function BellMatrix is defined in A264428.
BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
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MATHEMATICA
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Table[If[k <= n, Binomial[2n-2k, n-k] Binomial[2n-k-1, k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(2#-1)!!&, rows];
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PROG
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(Sage) # uses[bell_transform from A264428]
dblfact = a.list(n)
return bell_transform(n, dblfact)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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