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A163937 Triangle related to the o.g.f.s. of the right hand columns of A028421 (E(x,m=2,n)) 4
1, 1, 2, 2, 10, 3, 6, 52, 43, 4, 24, 308, 472, 136, 5, 120, 2088, 4980, 2832, 369, 6, 720, 16056, 53988, 49808, 13638, 918, 7, 5040, 138528, 616212, 826160, 381370, 57540, 2167, 8, 40320, 1327392, 7472952, 13570336, 9351260, 2469300, 222908, 4948, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The asymptotic expansions of the higher order exponential integral E(x,m=2,n) lead to triangle A028421, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A028421 have a nice structure Gf(p) = W2(z,p)/(1-z)^(2*p) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W2(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001147 (minus a(0)), see A163936 for more information.

LINKS

Table of n, a(n) for n=1..45.

FORMULA

a(n,m) = sum((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*stirling1(m+n-k-1,m-k),k=0..m-1)

EXAMPLE

The first few W2(z,p) polynomials are:

W2(z,p=1) = 1/(1-z)^2

W2(z,p=2) = (1+2*z)/(1-z)^4

W2(z,p=3) = (2+10*z+3*z^2)/(1-z)^6

W2(z,p=4) = (6+52*z+43*z^2+4*z^3)/(1-z)^8

MAPLE

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # [Johannes W. Meijer, revised Nov 27 2012]

CROSSREFS

Row sums equal A001147 (n>=1).

A000142, 2*A001705, are the first two left hand columns.

A000027 is the first right hand column.

Cf. A163931 (E(x,m,n)) and A028421.

Cf. A163936 (E(x,m=1,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

Sequence in context: A038036 A133631 A137450 * A083457 A163808 A223126

Adjacent sequences:  A163934 A163935 A163936 * A163938 A163939 A163940

KEYWORD

easy,nonn,tabl

AUTHOR

Johannes W. Meijer, Aug 13 2009

STATUS

approved

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Last modified November 28 21:04 EST 2014. Contains 250406 sequences.