A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

1, 3, 3, 11, 18, 6, 50, 105, 60, 10, 274, 675, 510, 150, 15, 1764, 4872, 4410, 1750, 315, 21, 13068, 39396, 40614, 19600, 4830, 588, 28, 109584, 354372, 403704, 224490, 68040, 11466, 1008, 36, 1026576, 3518100, 4342080, 2693250, 949095, 198450

Offset: 1

Author

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

Keywords

• easy
• nonn
• tabl

Comments

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).

Examples

			The first few rows of the triangle are:
[1]
[3, 3]
[11, 18, 6]
[50, 105, 60, 10]
		

Links

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

Crossrefs

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

Cf. A048994 (Stirling1), A000399 (row 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).

Programs

Maple

nmax:=8; with(combinat): for n1 from 1 to nmax do for m from 1 to n1 do a(n1, m) := (-1)^(n1+m)*binomial(m+1, 2)*stirling1(n1+1, m+1) od: od: seq(seq(a(n1,m), m=1..n1), n1=1..nmax);
# End program 1
with(combinat): 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+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: EA(x, 3, n);
# End program 2

Mathematica

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+1, 2] * StirlingS1[n+1, m+1]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 42]] (* Jean-François Alcover, Jun 01 2011, after formula *)

PARI

for(n=1,10, for(m=1,n, print1((-1)^(n+m)*binomial(m+1,2) *stirling(n+1,m+1,1), ", "))) \\ G. C. Greubel, Aug 08 2017

Formula

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

Extensions

Edited by Johannes W. Meijer, Sep 22 2012