1,2

The higher order exponential integrals E(x,m,n) are defined in A163931. The general formula for the asymptotic expansion E(x,m,n) ~ E(x,m-1,n+1)/x - n*E(x,m-1,n+2)/x^2 + n*(n+1) * E(x,m-1,n+3)/x^3 - n*(n+1)*(n+2)*E(x,m-1,n+4)/x^4 + .... , m => 1 and n => 1.

We used this formula and the asymptotic expansion of E(x,m=2,n), see A028421, to determine that E (x,m=3,n) ~ (exp(-x)/x^3)*(1 - (3+3*n)/x + (11+18*n+6*n^2)/x^2 - (50+105*n+ 60*n^2+ 10*n^3)/x^3 + .. ). This formula leads to the triangle coefficients given above.

The asymptotic expansion leads for the values of n from one to ten to known sequences, see the cross-references.

The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A001879, see A163938 for more information.

The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=3,n).

a(n,m) = (-1)^(n+m)*binomial(m+1,2)*stirling1(n+1,m+1) for n => 1 and 1<=m <= n .

The first few rows of the triangle are:

[1]

[3, 3]

[11, 18, 6]

[50, 105, 60, 10]

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

restart; with(combinat, stirling1): imax:=6; EA:=proc(x, m, n) local E, i; E:=0: for i from m-1 to imax+1 do E:=E + sum((-1)^(m+k+1)*binomial(k, m-1)*n^(k-m+1)* stirling1(i, k), k=m-1..i)/x^(i-m+1) od: E:= exp(-x)/x^(m)*E: return(E); end: EA(x, 3, n);

Cf. A163931 (E(x,m,n)) and A163938.

Cf. A048994 (Stirling1).

A000399 equals the rows sums.

A000254, 3*A000399, 6*A000454, 10*A000482, 15*A001233, 21*A001234 equal the first six left hand columns.

A000217, A006011 and A163933 equal the first three right hand columns.

The asymptotic expansion leads to A000399 (n=1), A001706 (n=2), A001712 (n=3), A001717 (n=4), A001722 (n=5), A051525 (n=6), A051546 (n=7), A051561 (n=8), A051563 (n=9) and A051565 (n=10).

Cf. A130534 (m=1), A028421 (m=2) and A163934 (m=4).

Sequence in context: A045495 A045494 A027416 this_sequence A007022 A011950 A124265

Adjacent sequences: A163929 A163930 A163931 this_sequence A163933 A163934 A163935

easy,nonn,tabl

Johannes W. Meijer & Nico Baken (meijgia(AT)hotmail.com and n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Oct 22 2009