|
| |
|
|
A163939
|
|
Triangle related to the o.g.f.s. of the right hand columns of A163934 (E(x,m=4,n))
|
|
6
| |
|
|
1, 6, 4, 35, 60, 10, 225, 690, 325, 20, 1624, 7588, 6762, 1316, 35, 13132, 85288, 120358, 46928, 4508, 56, 118124, 1004736, 2028660, 1298860, 265365, 13896, 84, 1172700, 12529400, 33896400, 31862400, 11077255, 1313610, 39915, 120
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| The asymptotic expansions of the higher order exponential integral E(x,m=4,n) lead to triangle A163934, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163934 have a nice structure Gf(p) = W4(z,p)/(1-z)^(2*p+2) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W4(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. The row sums of this triangle lead to A000457, see A163936 for more information.
|
|
|
FORMULA
| a(n,m) = sum((-1)^(n+k+1)*((m-k+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2,k)*stirling1(m+n-k+1,m-k+2),k=0..m-1)
|
|
|
EXAMPLE
| The first few W4(z,p) polynomials are:
W4(z,p=1) = 1/(1-z)^4
W4(z,p=2) = (6+4*z)/(1-z)^6
W4(z,p=3) = (35+60*z+10*z^2)/(1-z)^8
W4(z,p=4) = (225+690*z+325*z^2+20*z^3)/(1-z)^10
|
|
|
MAPLE
| with(combinat, stirling1): nmax:=8; for n from 1 to nmax do for m from 1 to n do a(n, m):=sum((-1)^(n+k+1)*((m-k+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2, k)*stirling1(m+n-k+1, m-k+2), k=0..m-1) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):=a(n, m): T:=T+1: od: od: seq(a(n), n=1..T-1);
|
|
|
CROSSREFS
| Row sums equal A000457.
A000399 equals the first left hand column.
A000292 equals the first right hand column.
Cf. A163931 (E(x,m,n)) and A163934.
Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163938 (E(x,m=3,n)).
Sequence in context: A121682 A191567 A163934 * A038258 A114330 A098657
Adjacent sequences: A163936 A163937 A163938 * A163940 A163941 A163942
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009
|
| |
|
|
