1,2

The asymptotic expansions of the higher order exponential integral E(x,m=3,n) lead to triangle A163932, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163932 have a nice structure Gf(p) = W3(z,p)/(1-z)^(2*p+1) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W3(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. The row sums of this triangle lead to A001879, see A163936 for more information.

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

The first few W3(z,p) polynomials are:

W3(z,p=1) = 1/(1-z)^3

W3(z,p=2) = (3+3*z)/(1-z)^5

W3(z,p=3) = (11+28*z+6*z^2)/(1-z)^7

W3(z,p=4) = (50+225*z+135*z^2+10*z^3)/(1-z)^9

with(combinat, stirling1): nmax:=8; 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)*(m-k)/2!)*binomial(2*n+1, k)*stirling1(m+n-k, m-k+1), 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 A001879.

A000254 equals the first left hand column.

A000217 equals the first right hand column.

Cf. A163931 (E(x,m,n)) and A163932.

Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163939 (E(x,m=4,n)).

Sequence in context: A007022 A011950 A124265 this_sequence A109937 A054101 A113892

Adjacent sequences: A163935 A163936 A163937 this_sequence A163939 A163940 A163941

easy,nonn,tabl

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