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A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).

%I A165674 #7 Jun 16 2016 23:27:40
%S A165674 1,3,1,11,5,1,50,26,7,1,274,154,47,9,1,1764,1044,342,74,11,1,13068,
%T A165674 8028,2754,638,107,13,1,109584,69264,24552,5944,1066,146,15,1,1026576,
%U A165674 663696,241128,60216,11274,1650,191,17,1
%N A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).
%C A165674 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.
%C A165674 The recurrence relations of the right hand columns of this triangle lead to Pascal's triangle A007318, their a(n) formulas 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.
%C A165674 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.
%C A165674 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 formulas, and the second one makes use of the asymptotic expansions of the E(x,m=2,n).
%C A165674 Amarnath Murthy discovered triangle A093905 which is the reversal of our triangle.
%C A165674 A165675 is an extended version of this triangle. Its reversal is A105954.
%C A165674 Triangle A094587 is generated by the asymptotic expansions of E(x,m=1,n).
%F A165674 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)!.
%F A165674 a(n,m) = product(i, i= m..n)*sum(1/i, i = m..n).
%p A165674 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: seq(seq(a(n, m), m = 1..n), n = 1..nmax);
%p A165674 # End program 1
%p A165674 nmax := nmax+1: m:=2; with(combinat): EA := proc(x, m, n) local E, i; E:=0: for i from m-1 to nmax+2 do E := E + sum((-1)^(m+k1+1) * binomial(k1, m-1) * n^(k1-m+1) * stirling1(i, k1), k1=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+2 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: seq(seq(a(n, m), m = 1..n),n = 1..nmax-1);
%p A165674 # End program 2
%p A165674 # Maple programs revised by _Johannes W. Meijer_, Sep 22 2012
%Y A165674 A093905 is the reversal of this triangle.
%Y A165674 A000254, A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562, A051564 are the first ten left hand columns.
%Y A165674 A080663, n>=2, is the third right hand column.
%Y A165674 A165676, A165677, A165678 and A165679 are the next right hand columns, A093344 gives the row sums.
%Y A165674 A073003 is Gompertz's constant.
%Y A165674 A094587 is generated by the asymptotic expansions of E(x, m=1, n).
%Y A165674 Cf. A165675, A105954 (Quet) and A067176 (Bottomley).
%Y A165674 Cf. A007318 (Pascal), A028421 (Wiggen), A126671 (Wood).
%K A165674 easy,nonn,tabl
%O A165674 1,2
%A A165674 _Johannes W. Meijer_, Oct 05 2009