%I A156919
%S A156919 1,2,1,4,10,1,8,60,36,1,16,296,516,116,1,32,1328,5168,3508,358,1,64,
%T A156919 5664,42960,64240,21120,1086,1,128,23488,320064,900560,660880,118632,
%U A156919 3272,1,256,95872,2225728
%N A156919 Table of coefficients of polynomials related to the Dirichlet eta function.
%C A156919 The SFn(z;m) formulae, see below, were discovered while studying certain 
               properties of the Dirichlet eta function.
%H A156919 D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting 
               Series Involving the Central Binomial Coefficient</a>, Am. Math. 
               Monthly 92 (1985) 449-457, Polynomial V in eq (17). [From R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Feb 24 2009]
%H A156919 Weisstein, Eric W., <a href="http://mathworld.wolfram.com/DirichletEtaFunction.html">
               Dirichlet Eta Function</a>
%F A156919 SF(z;n) = sum(m^(n-1)*4^(-m)*z^(m-1)*GAMMA(2*m+1)/(GAMMA(m)^2), m=1..infinity) 
               = P(z;n) / (2^(n+1)*(1-z)^((2*n+3)/2)) for n=0,1,2,3 . The polynomials 
               P(z;n) = sum( a(k)*z^k, k=0..n) generate the a(n) sequence.
%F A156919 If we write the sequence as a triangle the following relation holds: 
               DEF(n,m) = (2*m+2)*DEF(n-1,m) + (2*n-2*m+1)*DEF(n-1,m-1) with DEF(n,
               m=0) = 2^n and DEF(n,n) = 1. In view of the offset n=0,1,2,... and 
               m=0,1,..,n.
%e A156919 The first few rows of the "DEF" triangle are:
%e A156919 [1]
%e A156919 [2, 1]
%e A156919 [4, 10, 1]
%e A156919 [8, 60, 36, 1]
%e A156919 [16, 296, 516, 116, 1]
%e A156919 The first few P(z;n) are:
%e A156919 P(z; n=0) = 1
%e A156919 P(z; n=1) = 2 + z
%e A156919 P(z; n=2) = 4 + 10*z + z^2
%e A156919 P(z; n=3) = 8 + 60*z + 36*z^2 + z^3
%e A156919 The first few SF(z;n) are:
%e A156919 SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);
%e A156919 SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);
%e A156919 SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);
%e A156919 SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);
%p A156919 nmax:=8; mmax:=nmax: for m from 1 to mmax do DEF[0,m]:=0 end do: for 
               n from 0 to nmax do DEF[n,0]:=2^n end do: for n from 1 to nmax do 
               for m from 1 to mmax do DEF[n,m]:=(2*m+2)*DEF[n-1,m] + (2*n-2*m+1)*DEF[n-1,
               m-1] end do end do: for n from 0 to nmax do for m from 0 to n do 
               DEF[n*(n+1)/2+m]:=DEF[n,m] end do end do: a:=n-> DEF[n]: seq(a(n), 
               n=0..((1/2)*nmax^2+(3/2)*nmax));
%Y A156919 A142963 and this sequence can be mapped onto the A156920 triangle.
%Y A156919 FP1 sequences A000340, A156922, A156923, A156924.
%Y A156919 FP2 sequences A050488(n+1), A142965, A142966, A142968.
%Y A156919 Row sums (n) = A001147(n+1)
%Y A156919 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 12 
               2009: (Start)
%Y A156919 Appears in A162005, A000182, A162006 and A162007.
%Y A156919 (End)
%Y A156919 Sequence in context: A137634 A100229 A071949 this_sequence A038195 A038521 
               A134654
%Y A156919 Adjacent sequences: A156916 A156917 A156918 this_sequence A156920 A156921 
               A156922
%K A156919 easy,nonn,tabl
%O A156919 0,2
%A A156919 Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009, Jun 24 2009