%I #28 Jun 07 2023 07:53:12
%S 1,6,4,35,40,10,225,340,150,20,1624,2940,1750,420,35,13132,27076,
%T 19600,6440,980,56,118124,269136,224490,90720,19110,2016,84,1172700,
%U 2894720,2693250,1265460,330750,48720,3780,120
%N Triangle related to the asymptotic expansion of E(x,m=4,n).
%C The higher order exponential integrals E(x,m,n) are defined in A163931 while the general formula for their asymptotic expansion can be found in A163932.
%C We used the latter formula and the asymptotic expansion of E(x,m=3,n), see A163932, to determine that E(x,m=4,n) ~ (exp(-x)/x^4)*(1 - (6+4*n)/x + (35+40*n+ 10*n^2)/x^2 - (225+340*n+ 150*n^2+20*n^3)/x^3 + ... ). This formula leads to the triangle coefficients given above.
%C The asymptotic expansion leads for the values of n from one to five to known sequences, see the cross-references.
%C The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A000457, see A163939 for more information.
%C The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).
%H G. C. Greubel, <a href="/A163934/b163934.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n.
%e The first few rows of the triangle are:
%e 1;
%e 6, 4;
%e 35, 40, 10;
%e 225, 340, 150, 20;
%p with(combinat): A163934 := proc(n,m): (-1)^(n+m)* binomial(m+2, 3) *stirling1(n+2, m+2) end: seq(seq(A163934(n,m), m=1..n), n=1..8);
%p with(combinat): imax:=6; EA:=proc(x,m,n) local E, i; E:=0: for i from m-1 to imax+2 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,4,n);
%p # Maple programs revised by _Johannes W. Meijer_, Sep 11 2012
%t a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+2, 3] * StirlingS1[n+2, m+2]; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* _Jean-François Alcover_, Jun 01 2011, after formula *)
%Y Cf. A163931 (E(x,m,n)), A163932 and A163939.
%Y Cf. A048994 (Stirling1), A000454 (row sums).
%Y A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.
%Y A000292, A027777 and A163935 equal the first three right hand columns.
%Y The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).
%Y Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).
%K easy,nonn,tabl
%O 1,2
%A _Johannes W. Meijer_, Aug 13 2009