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 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
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*Stirling1(m+n-k-1,m-k), 1 <= m <= n.
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
MATHEMATICA
Table[Sum[(-1)^(n + k + 1)*((m - k)/1!)*Binomial[2*n, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)
PROG
(PARI) for(n=1, 10, for(m=1, n, print1(sum(k=0, m-1, (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k) *stirling1(m+n-k-1, m-k)), ", "))) \\ G. C. Greubel, Aug 13 2017
KEYWORD
AUTHOR
Johannes W. Meijer, Aug 13 2009
STATUS
approved