A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

1, 6, 4, 35, 40, 10, 225, 340, 150, 20, 1624, 2940, 1750, 420, 35, 13132, 27076, 19600, 6440, 980, 56, 118124, 269136, 224490, 90720, 19110, 2016, 84, 1172700, 2894720, 2693250, 1265460, 330750, 48720, 3780, 120

Offset: 1

Author

Johannes W. Meijer, Aug 13 2009

Keywords

• easy
• nonn
• tabl

Comments

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.

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.

The asymptotic expansion leads for the values of n from one to five 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 A000457, see A163939 for more information.

The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).

Examples

			The first few rows of the triangle are:
1;
6, 4;
35, 40, 10;
225, 340, 150, 20;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A163931 (E(x,m,n)), A163932 and A163939.

Cf. A048994 (Stirling1), A000454 (row sums).

A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.

A000292, A027777 and A163935 equal the first three right hand columns.

The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).

Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).

Programs

Maple

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);
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);
# Maple programs revised by Johannes W. Meijer, Sep 11 2012

Mathematica

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 *)

Formula

a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n.