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A028421
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Array of numbers f(n,k) from n-th differences of sequence {1/x^2}; n-th difference is n!*P(x)/(D^2) where P(x) is a degree-n polynomial: P(n) = Sum_k { f(n,k)*x^k } and D = x(x+1) ...(x+n-1)(x+n).
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26
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1, 1, 2, 2, 6, 3, 6, 22, 18, 4, 24, 100, 105, 40, 5, 120, 548, 675, 340, 75, 6, 720, 3528, 4872, 2940, 875, 126, 7, 5040, 26136, 39396, 27076, 9800, 1932, 196, 8, 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9, 362880, 2053152
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
The higher order exponential integrals E(x,m,n) are defined in A163931 and the general formula of the asymptotic expansion of E(x,m,n) can be found in A163932.
We used the general formula and the asymptotic expansion of E(x,m=1,n), see A130534, to determine that E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2+6*n+3*n^2)/x^2 - (6 + 22*n + 18*n^2+ 4*n^3)/x^3 + ... ) which can be verified with the EA(x,2,n) formula, see A163932. The coefficients in the denominators of this expansion lead to the sequence given above.
The asymptotic expansion of E(x,m=2,n) leads for n from one to ten to known sequences, see the cross-references. With these sequences one can form the triangles A165674 (left hand columns) and A093905 (right hand columns).
(End)
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FORMULA
| f(n, k) = (k+1)*s[ n+1, k+1 ] (for n >= k) where s[ n, k ] is an unsigned stirling number of the first kind.
E.g.f.: -ln(1-x)/(1-x)^y. - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 12 2003
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MAPLE
| If we shift the offset from 0 to 1 we get the expression f(n-1, m-1) = a(n, m) = (-1)^(n+m) * (m) * stirling1(n, m)/1! for the sequence given above, see the Maple program.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
nmax:=9; 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, 1)*stirling1(n, m); f(n-1, m-1):= a(n, m): od: od: T:=0: for n from 0 to nmax-1 do for m from 0 to n do a(T):=f(n, m); T:=T+1: od: od: seq(a(n), n=0..T-1);
(End)
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MATHEMATICA
| f[n_, k_] = (k + 1) StirlingS1[n + 1, k + 1] // Abs; Flatten[Table[f[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* From Jean-François Alcover, Jun 1 2011, after formula *)
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CROSSREFS
| Row sums give A000254.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)
A000142, A052517, 3*A000399, 5*A000482 are the first four left hand columns,
A000027, A002411 are the first two right hand columns.
The asymptotic expansion of E(x,m=2,n) leads to A000254 (n=1), A001705 (n=2), A001711 (n=3), A001716 (n=4), A001721 (n=5), A051524 (n=6), A051545 (n=7), A051560 (n=8), A051562 (n=9), A051564 (n=10), A093905 (triangle) and A165674 (triangle).
Cf. A163931 (E(x,m,n)), A130534 (m=1), A163932 (m=3), A163934 (m=4), A074246 (E(x,m=2,n+1)).
(End)
Sequence in context: A174833 A085738 A100641 * A081745 A129889 A130712
Adjacent sequences: A028418 A028419 A028420 * A028422 A028423 A028424
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KEYWORD
| tabl,nonn
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AUTHOR
| Peter Wiggen (wiggen(AT)math.psu.edu)
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