|
| |
|
|
A163938
|
|
Triangle related to the o.g.f.s. of the right hand columns of A163932 (E(x,m=3,n))
|
|
6
| |
|
|
1, 3, 3, 11, 28, 6, 50, 225, 135, 10, 274, 1858, 2092, 486, 15, 1764, 16464, 29148, 13482, 1491, 21, 13068, 158352, 398640, 301220, 70485, 4152, 28, 109584, 1655172, 5552724, 6132780, 2432070, 322971, 10863, 36
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| The asymptotic expansions of the higher order exponential integral E(x,m=3,n) lead to triangle A163932, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163932 have a nice structure Gf(p) = W3(z,p)/(1-z)^(2*p+1) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W3(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. The row sums of this triangle lead to A001879, see A163936 for more information.
|
|
|
FORMULA
| a(n,m) = sum((-1)^(n+k+1)*((m-k+1)*(m-k)/2!)*binomial(2*n+1,k)*stirling1(m+n-k,m-k+1),k=0..m-1)
|
|
|
EXAMPLE
| The first few W3(z,p) polynomials are:
W3(z,p=1) = 1/(1-z)^3
W3(z,p=2) = (3+3*z)/(1-z)^5
W3(z,p=3) = (11+28*z+6*z^2)/(1-z)^7
W3(z,p=4) = (50+225*z+135*z^2+10*z^3)/(1-z)^9
|
|
|
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+1)*(m-k)/2!)*binomial(2*n+1, k)*stirling1(m+n-k, m-k+1), 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 A001879.
A000254 equals the first left hand column.
A000217 equals the first right hand column.
Cf. A163931 (E(x,m,n)) and A163932.
Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163939 (E(x,m=4,n)).
Sequence in context: A007022 A011950 A124265 * A109937 A054101 A176956
Adjacent sequences: A163935 A163936 A163937 * A163939 A163940 A163941
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009
|
| |
|
|
