%I A039683
%S A039683 1,2,1,8,6,1,48,44,12,1,384,400,140,20,1,3840,4384,1800,340,30,1,46080,
%T A039683 56448,25984,5880,700,42,1,645120,836352,420224,108304,15680,1288,56,1,
%U A039683 10321920,14026752,7559936,2153088,359184,36288,2184,72,1
%V A039683 1,-2,1,8,-6,1,-48,44,-12,1,384,-400,140,-20,1,-3840,4384,-1800,340,-30,
               1,
%W A039683 46080,-56448,25984,-5880,700,-42,1,-645120,836352,-420224,108304,-15680,
%X A039683 1288,-56,1,10321920,-14026752,7559936,-2153088,359184,-36288,2184,-72,
               1
%N A039683 Double Pochhammer triangle: expansion of x(x+2)(x+4)..(x+2n-2).
%C A039683 a(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
%C A039683 Exponential Riordan array [1/(1+2x),ln(1+2x)/2]. The unsigned triangle 
               is [1/(1-2x),ln(1/sqrt(1-2x))]. [From Paul Barry (pbarry(AT)wit.ie), 
               Apr 29 2009]
%D A039683 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.
%H A039683 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A039683.text">
               First 9 rows and comment</a>.
%F A039683 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!.
%F A039683 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.rs), 
               Feb 11 2003
%F A039683 a(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 
               triangle).
%e A039683 {1}, {2,1}, {8,6,1}, {48,44,12,1}, ...
%e A039683 Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 29 2009: (Start)
%e A039683 The unsigned triangle [1/(1-2x),ln(1/sqrt(1-2x))] has production matrix
%e A039683 2, 1,
%e A039683 4, 4, 1,
%e A039683 8, 12, 6, 1,
%e A039683 16, 32, 24, 8, 1,
%e A039683 32, 80, 80, 40, 10, 1,
%e A039683 64, 192, 240, 160, 60, 12, 1
%e A039683 which is A007318^{2} beheaded. (End)
%t A039683 Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 
               6} ]
%Y A039683 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.
%Y A039683 Sequence in context: A008517 A142336 A114193 this_sequence A108084 A108085 
               A154897
%Y A039683 Adjacent sequences: A039680 A039681 A039682 this_sequence A039684 A039685 
               A039686
%K A039683 sign,tabl
%O A039683 1,2
%A A039683 wouter.meeussen(AT)pandora.be
%E A039683 Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).