Search: id:A008296
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%I A008296
%S A008296 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,
%T A008296 196,49,0,154,28,1,5040,1368,944,0,231,252,294,36,1,40320,11016,5340,820,
               1365,
%U A008296 987,1050,510,45,1,362880,98208,34716,9020,7645,3003,1617,2970,825,55,
               1,3628800
%V A008296 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,
%W A008296 -196,49,0,154,28,1,-5040,-1368,944,0,-231,252,294,36,1,40320,11016,-5340,
               -820,1365,
%X A008296 -987,1050,510,45,1,-362880,-98208,34716,9020,-7645,3003,-1617,2970,825,
               55,1,3628800
%N A008296 Triangle of Lehmer-Comtet numbers of first kind.
%C A008296 Triangle arising in expansion of ((1+x)log(1+x))^n.
%D A008296 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.
%D A008296 D. H. Lehmer, "Numbers Associated with Stirling Numbers and x^x", Rocky 
               Mountain J. Math., 15(2) 1985, pp. 461-475.
%F A008296 E.g.f. for a(n, k): (1/k!)[ (1+x)*ln(1+x) ]^k. - Leonard Smiley (smiley(AT)math.uaa.alaska.edu)
%F A008296 Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
%F A008296 a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
%F A008296 a(n, k) = Sum_{l} binomial(l, k)*k^(l-k)*stirling1(n, l).
%e A008296 Triangle begins:
%e A008296 1;
%e A008296 1,1;
%e A008296 -1,3,1;
%e A008296 2,-1,6,1;
%e A008296 -6,0,5,10,1;
%e A008296 24,4,-15,25,15,1;
%e A008296 ...
%p A008296 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:
%o A008296 (PARI) T(n,k)=if(k<1|k>n,0,n!*polcoeff(((1+x)*log(1+x+x*O(x^n)))^k/k!,
               n))
%Y A008296 Cf. A039621.
%Y A008296 Diagonals give A000142, A045406, A000217, A059302. Row sums give A005727.
%Y A008296 Sequence in context: A131918 A010123 A039620 this_sequence A140185 A106790 
               A078897
%Y A008296 Adjacent sequences: A008293 A008294 A008295 this_sequence A008297 A008298 
               A008299
%K A008296 sign,tabl,easy,nice
%O A008296 1,5
%A A008296 N. J. A. Sloane (njas(AT)research.att.com).
%E A008296 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 26 2001
%E A008296 Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion 
               of Andrew Robbins, Dec 11 2007

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