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A163934
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Triangle related to the asymptotic expansion of E(x,m=4,n)
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11
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1, 6, 4, 35, 40, 10, 225, 340, 150, 20, 1624, 2940, 1750, 420, 35, 13132, 27076, 19600, 6440, 980, 56, 118124, 269136, 224490, 90720, 19110, 2016, 84, 1172700, 2894720, 2693250, 1265460, 330750, 48720, 3780, 120
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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.
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 + ... ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to five to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A000457, see A163939 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).
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FORMULA
| a(n,m) = (-1)^(n+m)*binomial(m+2,3)*stirling1(n+2,m+2) for n => 1 and 1<=m <= n .
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EXAMPLE
| The first few rows of the triangle are:
[1]
[6, 4]
[35, 40, 10]
[225, 340,150, 20]
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MAPLE
| restart; nmax:=8; mmax:=nmax: with(combinat, stirling1): for n from 1 to nmax do for m from 1 to n do a(n, m):=(-1)^(n+m)* binomial(m+2, 3)*stirling1(n+2, m+2) od; od: T:=0: for n from 1 to nmax do for m from 1 to n do a(T):=a(n, m); T:=T+1: od: od: seq(a(n), n=0..T-1);
restart; with(combinat, stirling1): 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);
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MATHEMATICA
| 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]] (* From Jean-François Alcover, Jun 1 2011, after formula *)
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CROSSREFS
| Cf. A163931 (E(x,m,n)), A163932 and A163939.
Cf. A048994 (Stirling1).
A000454 equals the rows sums.
A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.
A000292, A027777 and A163935 equal the first three right hand columns.
The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).
Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).
Sequence in context: A133837 A121682 A191567 * A163939 A038258 A114330
Adjacent sequences: A163931 A163932 A163933 * A163935 A163936 A163937
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009
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