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A008296
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Triangle of Lehmer-Comtet numbers of first kind.
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5
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1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Triangle arising in expansion of ((1+x)log(1+x))^n.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.
D. H. Lehmer, "Numbers Associated with Stirling Numbers and x^x", Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
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FORMULA
| E.g.f. for a(n, k): (1/k!)[ (1+x)*ln(1+x) ]^k. - Leonard Smiley (smiley(AT)math.uaa.alaska.edu)
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{l} binomial(l, k)*k^(l-k)*stirling1(n, l).
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EXAMPLE
| Triangle begins:
1;
1,1;
-1,3,1;
2,-1,6,1;
-6,0,5,10,1;
24,4,-15,25,15,1;
...
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MAPLE
| with(combinat): for n from 1 to 20 do for k from 1 to n do printf(`%d, `, sum(binomial(l, k)*k^(l-k)*stirling1(n, l), l=k..n)) od: od:
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MATHEMATICA
| a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1, k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* From Jean-François Alcover, Apr 29 2011 *)
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PROG
| (PARI) T(n, k)=if(k<1|k>n, 0, n!*polcoeff(((1+x)*log(1+x+x*O(x^n)))^k/k!, n))
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CROSSREFS
| Cf. A039621.
Diagonals give A000142, A045406, A000217, A059302. Row sums give A005727.
Sequence in context: A131918 A010123 A039620 * A140185 A106790 A078897
Adjacent sequences: A008293 A008294 A008295 * A008297 A008298 A008299
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KEYWORD
| sign,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 26 2001
Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Robbins, Dec 11 2007
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