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%I #79 Jun 11 2022 03:45:53
%S 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,
%T 70,21,1,720,188,-196,49,0,154,28,1,-5040,-1368,944,0,-231,252,294,36,
%U 1,40320,11016,-5340,-820,1365,-987,1050,510,45,1,-362880,-98208,34716,9020,-7645,3003,-1617,2970,825,55,1,3628800
%N Triangle of Lehmer-Comtet numbers of the first kind.
%C Triangle arising in the expansion of ((1+x)*log(1+x))^n.
%C Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 16 2016
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.
%H Alois P. Heinz, <a href="/A008296/b008296.txt">Rows n = 1..141, flattened</a>
%H H. W. Gould, <a href="http://dx.doi.org/10.1216/rmjm/1181072076">A Set of Polynomials Associated with the Higher Derivatives of y = x^x</a>, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
%H Tian-Xiao He and Yuanziyi Zhang, <a href="https://arxiv.org/abs/2105.07262">Centralizers of the Riordan Group</a>, arXiv:2105.07262 [math.CO], 2021.
%H D. H. Lehmer, <a href="https://projecteuclid.org/euclid.rmjm/1250127221">Numbers Associated with Stirling Numbers and x^x</a>, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
%F E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - _Len Smiley_
%F Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
%F a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
%F a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
%F From _Peter Bala_, Mar 14 2012: (Start)
%F E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
%F (End)
%F Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - _Alois P. Heinz_, Aug 25 2021
%e Triangle begins:
%e 1;
%e 1, 1;
%e -1, 3, 1;
%e 2, -1, 6, 1;
%e -6, 0, 5, 10, 1;
%e 24, 4, -15, 25, 15, 1;
%e ...
%p for n from 1 to 20 do for k from 1 to n do
%p printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
%p # second program:
%p A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
%p seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # _Mélika Tebni_, Aug 22 2021
%t a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
%t 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]]
%t (* _Jean-François Alcover_, Apr 29 2011 *)
%o (PARI) {T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* _Michael Somos_, Nov 15 2002 */
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
%o bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # _Peter Luschny_, Jan 16 2016
%Y Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
%Y Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
%Y Diagonals: A000142, A000217, A059302.
%Y Cf. A176118.
%K sign,tabl,easy,nice
%O 1,5
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Jan 26 2001
%E Edited by _N. J. A. Sloane_ at the suggestion of Andrew Robbins, Dec 11 2007
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