1,3

The asymptotic expansions of the higher order exponential integral E(x,m=2,n) lead to triangle A028421, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A028421 have a nice structure Gf(p) = W2(z,p)/(1-z)^(2*p) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W2(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. The row sums of this triangle lead to A001147 (minus a(0)), see A163936 for more information.

a(n,m) = sum((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*stirling1(m+n-k-1,m-k),k=0..m-1)

The first few W2(z,p) polynomials are:

W2(z,p=1) = 1/(1-z)^2

W2(z,p=2) = (1+2*z)/(1-z)^4

W2(z,p=3) = (2+10*z+3*z^2)/(1-z)^6

W2(z,p=4) = (6+52*z+43*z^2+4*z^3)/(1-z)^8

with(combinat, stirling1): nmax:=9; for n from 1 to nmax do for m from 1 to n do a(n, m):=sum((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k)*stirling1(m+n-k-1, m-k), k=0..m-1) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):=a(n, m): T:=T+1: od: od: seq(a(n), n=1..T-1);

Row sums equal A001147 (n=>1).

A000142, 2*A001705, are the first two left hand columns.

A000027 is the first right hand column.

Cf. A163931 (E(x,m,n)) and A028421.

Cf. A163936 (E(x,m=1,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

Sequence in context: A038036 A133631 A137450 this_sequence A083457 A163808 A127058

Adjacent sequences: A163934 A163935 A163936 this_sequence A163938 A163939 A163940

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009