来自在线整数百科全书的问候语!http://oeis.org/ Search: id:a163934 Showing 1-1 of 1 %I A163934 %S A163934 1,6,4,35,40,10,225,340,150,20,1624,2940,1750,420,35,13132,27076, %T A163934 19600,6440,980,56,118124,269136,224490,90720,19110,2016,84,1172700, %U A163934 2894720,2693250,1265460,330750,48720,3780,120 %N A163934 Triangle related to the asymptotic expansion of E(x,m=4,n). %C A163934 The higher order exponential integrals E(x,m,n) are defined in A163931 while the general formula for their asymptotic expansion can be found in A163932. %C A163934 We used the latter formula and the asymptotic expansion of E(x,m=3,n), see A163932, to determine that E(x,m=4,n) ~ (exp(-x)/x^4)*(1 - (6+4*n)/x + (35+40*n+ 10*n^2)/x^2 - (225+340*n+ 150*n^2+20*n^3)/x^3 + ... )这个公式导致了上面给出的三角形系数。%C.A1639 34,将n的值从1到五引向已知序列,参见交叉引用。{%C A1639 34,这个三角形的右手列的O.G.F.S的分子为z=1到A000 047,见A1639 39以获得更多的信息。n,a(n)的前50行,肥育的表%F A163934 a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n. %e A163934 The first few rows of the triangle are: %e A163934 1; %e A163934 6, 4; %e A163934 35, 40, 10; %e A163934 225, 340, 150, 20; %p A163934 with(combinat): A163934 := proc(n,m): (-1)^(n+m)* binomial(m+2, 3) *stirling1(n+2, m+2) end: seq(seq(A163934(n,m), m=1..n), n=1..8); %p A163934 with(combinat): imax:=6; EA:=proc(x,m,n) local E, i; E:=0: for i from m-1 to imax+2 do E:=E + sum((-1)^(m+k+1)*binomial(k,m-1)*n^(k-m+1)* stirling1(i, k), k=m-1..i)/x^(i-m+1) od: E:= exp(-x)/x^(m)*E: return(E); end: EA(x,4,n); %p A163934 # Maple programs revised by _Johannes W. Meijer_, Sep 11 2012 %t A163934 a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+2, 3] * StirlingS1[n+2, m+2]; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* _Jean-François Alcover_, Jun 01 2011, after formula *) %Y A163934 Cf. A163931 (E(x,m,n)), A163932 and A163939. %Y A163934 Cf. A048994 (Stirling1), A000454 (row sums). %Y A163934 A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns. %Y A163934 A000292, A027777 and A163935 equal the first three right hand columns. %Y A163934 The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5). %Y A163934 Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3). %K A163934 easy,nonn,tabl %O A163934 1,2 %A A163934 _Johannes W. Meijer_, Aug 13 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE