0,2

The SFn(z;m) formulae, see below, were discovered while studying certain properties of the Dirichlet eta function.

D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457, Polynomial V in eq (17). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 24 2009]

Weisstein, Eric W., Dirichlet Eta Function

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.

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.

The first few rows of the "DEF" triangle are:

[1]

[2, 1]

[4, 10, 1]

[8, 60, 36, 1]

[16, 296, 516, 116, 1]

The first few P(z;n) are:

P(z; n=0) = 1

P(z; n=1) = 2 + z

P(z; n=2) = 4 + 10*z + z^2

P(z; n=3) = 8 + 60*z + 36*z^2 + z^3

The first few SF(z;n) are:

SF(z; n=0) = (1/2)*(1)/(1-z)^(3/2);

SF(z; n=1) = (1/4)*(2+z)/(1-z)^(5/2);

SF(z; n=2) = (1/8)*(4+10*z+z^2)/(1-z)^(7/2);

SF(z; n=3) = (1/16)*(8+60*z+36*z^2+z^3)/(1-z)^(9/2);

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));

A142963 and this sequence can be mapped onto the A156920 triangle.

FP1 sequences A000340, A156922, A156923, A156924.

FP2 sequences A050488(n+1), A142965, A142966, A142968.

Row sums (n) = A001147(n+1)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 12 2009: (Start)

Appears in A162005, A000182, A162006 and A162007.

(End)

Sequence in context: A137634 A100229 A071949 this_sequence A038195 A038521 A134654

Adjacent sequences: A156916 A156917 A156918 this_sequence A156920 A156921 A156922

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009, Jun 24 2009