1,2

The asymptotic expansions of the higher order exponential integral E(x,m=4,n) lead to triangle A163934, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163934 have a nice structure Gf(p) = W4(z,p)/(1-z)^(2*p+2) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W4(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. The row sums of this triangle lead to A000457, see A163936 for more information.

a(n,m) = sum((-1)^(n+k+1)*((m-k+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2,k)*stirling1(m+n-k+1,m-k+2),k=0..m-1)

The first few W4(z,p) polynomials are:

W4(z,p=1) = 1/(1-z)^4

W4(z,p=2) = (6+4*z)/(1-z)^6

W4(z,p=3) = (35+60*z+10*z^2)/(1-z)^8

W4(z,p=4) = (225+690*z+325*z^2+20*z^3)/(1-z)^10

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+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2, k)*stirling1(m+n-k+1, m-k+2), 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 A000457.

A000399 equals the first left hand column.

A000292 equals the first right hand column.

Cf. A163931 (E(x,m,n)) and A163934.

Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163938 (E(x,m=3,n)).

Sequence in context: A133837 A121682 A163934 this_sequence A038258 A114330 A098657

Adjacent sequences: A163936 A163937 A163938 this_sequence A163940 A163941 A163942

easy,nonn,tabl

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