|
%I #10 Aug 13 2017 20:55:38
%S 1,6,4,35,60,10,225,690,325,20,1624,7588,6762,1316,35,13132,85288,
%T 120358,46928,4508,56,118124,1004736,2028660,1298860,265365,13896,84,
%U 1172700,12529400,33896400,31862400,11077255,1313610,39915,120
%N Triangle related to the o.g.f.s. of the right hand columns of A163934 (E(x,m=4,n)).
%C 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.
%H G. C. Greubel, <a href="/A163939/b163939.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+2,3)* binomial(2*n+2,k)*stirling1(m+n-k+1,m-k+2), for 1<= m <=n.
%e The first few W4(z,p) polynomials are:
%e W4(z,p=1) = 1/(1-z)^4
%e W4(z,p=2) = (6+4*z)/(1-z)^6
%e W4(z,p=3) = (35+60*z+10*z^2)/(1-z)^8
%e W4(z,p=4) = (225+690*z+325*z^2+20*z^3)/(1-z)^10
%p with(combinat): a := proc(n, m): add((-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) 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 + 2, 3]*Binomial[2*n + 2, k]*StirlingS1[m + n - k + 1, m - k + 2], {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+2,3)* binomial(2*n+2,k)*stirling(m+n-k+1,m-k+2,1)), ", "))) \\ _G. C. Greubel_, Aug 13 2017
%Y Row sums equal A000457.
%Y A000399 equals the first left hand column.
%Y A000292 equals the first right hand column.
%Y Cf. A163931 (E(x,m,n)) and A163934.
%Y Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163938 (E(x,m=3,n)).
%K easy,nonn,tabl
%O 1,2
%A _Johannes W. Meijer_, Aug 13 2009
|
