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%I #11 Aug 14 2017 03:00:37
%S 1,3,3,11,28,6,50,225,135,10,274,1858,2092,486,15,1764,16464,29148,
%T 13482,1491,21,13068,158352,398640,301220,70485,4152,28,109584,
%U 1655172,5552724,6132780,2432070,322971,10863,36
%N Triangle related to the o.g.f.s. of the right hand columns of A163932 (E(x, m=3, n)).
%C 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.
%H G. C. Greubel, <a href="/A163938/b163938.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*binomial(m-k+1,2) *binomial(2*n+1,k) *stirling1(m+n-k,m-k+1), for 1 <= m <= n.
%e The first few W3(z,p) polynomials are:
%e W3(z,p=1) = 1/(1-z)^3
%e W3(z,p=2) = (3 + 3*z)/(1-z)^5
%e W3(z,p=3) = (11 + 28*z + 6*z^2)/(1-z)^7
%e W3(z,p=4) = (50 + 225*z + 135*z^2 + 10*z^3)/(1-z)^9
%p with(combinat): a := proc(n, m): add((-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) end: seq(seq(a(n, m), m=1..n), n=1..8); # _Johannes W. Meijer_, revised Nov 27 2012
%t Table[Sum[(-1)^(n + k + 1)*Binomial[m - k + 1, 2]*Binomial[2*n + 1, k]*StirlingS1[m + n - k, m - k + 1], {k, 0, m - 1}], {n, 1, 50}, {m, 1, n}] // Flatten (* _G. C. Greubel_, Aug 13 2017 *)
%o (PARI) for(n=1,10, for(m=1,n, print1(sum(k=0,m-1, (-1)^(n+k+1)* binomial(m-k+1,2)*binomial(2*n+1,k) *stirling(m+n-k,m-k+1, 1)) ,", "))) \\ _G. C. Greubel_, Aug 13 2017
%Y Row sums equal A001879.
%Y A000254 equals the first left hand column.
%Y A000217 equals the first right hand column.
%Y Cf. A163931 (E(x,m,n)) and A163932.
%Y Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163939 (E(x,m=4,n)).
%K easy,nonn,tabl
%O 1,2
%A _Johannes W. Meijer_, Aug 13 2009
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