1,2

The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2+6*n+3*n^2)/x^2 - (6+22*n+18*n^2+ 4*n^3)/x^3 + ... ) is discussed in A028421. The formula for the asymptotic expansion leads for n = 1, 2, 3, .. , to the left hand columns of the triangle given above.

The recurrence relations of the right hand columns of this triangle lead to Pascal's triangle A007318, their a(n) formulae lead to Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; cf. A080663, A165676, A165677, A165678 and A165679.

The row sums of this triangle lead to A093344. Surprisingly the e.g.f. of the row sums Egf(x) = (exp(1)*Ei(1,1-x) - exp(1)*Ei(1,1))/(1-x) leads to the exponential integrals in view of the fact that E(x,m=1,n=1) = Ei(n=1,x). We point out that exp(1)*Ei(1,1) = A073003.

The Maple programs generate the coefficients of the triangle given above. The first one makes use of a relation between the triangle coefficients, see the formulae, and the second one makes use of the asymptotic expansions of the E(x,m=2,n).

Amarnath Murthy discovered triangle A093905 which is the reversal of our triangle.

A165675 is an extended version of this triangle. Its reversal is A105954.

Triangle A094587 is generated by the asymptotic expansions of E(x,m=1,n).

a(n,m) = (n-m+1)*a(n-1,m) + a(n-1,m-1), for 2 =< m =< n-1, with a(n,n) = 1 and a(n,1) = n*a(n-1,1) + (n-1)!

a(n,m) = product(i, i= m..n)*sum(1/i, i = m..n)

restart; nmax:=9; for n from 1 to nmax do a(n, n):=1 od: for n from 2 to nmax do a(n, 1) := n*a(n-1, 1)+ (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do a(n, m):=(n-m+1)*a(n-1, m)+a(n-1, m-1) od: od: T:=0: for n from 1 to nmax do for m from 1 to n do T:=T+1: a(T):=a(n, m); od: od: seq(a(n), n=1..T);

restart; m:=2; nmax:=10; with(combinat, stirling1): EA:=proc(x, m, n) local E, i; E:=0: for i from m-1 to nmax+1+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: for n1 from 1 to nmax do f(n1-1):=simplify(exp(x)*x^(nmax+3)*EA(x, m, n1)); for m1 from 0 to nmax+1+1 do b(n1-1, m1):=coeff(f(n1-1), x, nmax+2-m1) od: od: for n1 from 0 to nmax-1 do for m1 from 0 to n1-m+1 do a(n1-m+2, m1+1):= abs(b(m1, n1-m1)) od: od: T:=0: for n1 from 1 to nmax-m+1 do for m1 from 1 to n1 do T:=T+1: a(T):=a(n1, m1); od: od: seq(a(n1), n1=1..T);

A093905 is the reversal of this triangle.

A000254, A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562, A051564 are the first ten left hand columns.

A080663, n=>2, is the third right hand column.

A165676, A165677, A165678 and A165679 are the next right hand columns, A093344 gives the row sums.

A073003 is Gompertz's constant.

A094587 is generated by the asymptotic expansions of E(x, m=1, n).

Cf. A165675, A105954 (Quet) and A067176 (Bottomley).

Cf. A007318 (Pascal), A028421 (Wiggen), A126671 (Wood).

Sequence in context: A120291 A099001 A119947 this_sequence A027446 A027516 A092808

Adjacent sequences: A165671 A165672 A165673 this_sequence A165675 A165676 A165677

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 05 2009