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%I #30 Jan 21 2020 00:05:10
%S 1,1,1,1,3,1,1,6,8,1,1,10,29,24,1,1,15,75,145,89,1,1,21,160,545,814,
%T 415,1,1,28,301,1575,4179,5243,2372,1,1,36,518,3836,15659,34860,38618,
%U 16072,1,1,45,834,8274,47775,163191,318926,321690,125673,1,1,55,1275,16290
%N Coefficients of a special case of Poisson-Charlier polynomials.
%C Diagonals: A000012, A000217; A000012, A002104. - _Philippe Deléham_, Jun 12 2004
%C The sequence a(n) = Sum_{k = 0..n} T(n,k)*x^(n-k) is the binomial transform of the sequence b(n) = (n+x-1)! / (x-1)!. - _Philippe Deléham_, Jun 18 2004
%H E. A. Enneking and J. C. Ahuja, <a href="http://www.fq.math.ca/Scanned/14-1/enneking.pdf">Generalized Bell numbers</a>, Fib. Quart., 14 (1976), 67-73.
%H C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
%F Reference gives a recurrence.
%F Sum_{k = 0..n} T(n, k)*x^(n-k) = A000522(n), A001339(n), A082030(n) for x = 1, 2, 3 respectively. Sum_{k = 0..n} T(n, k)*2^k = A081367(n). - _Philippe Deléham_, Jun 12 2004
%F Let P(x, n) = Sum_{k = 0..n} T(n, k)*x^k, then Sum_{n>=0} P(x, n)*t^n / n! = exp(xt)/(1-xt)^(1/x). - _Philippe Deléham_, Jun 12 2004
%F T(n, 0) = 1, T(n, k) = (-1)^k * Sum_{i=n-k..n} (-1)^i*C(n, i)*S1(i, n-k), where S1 = Stirling numbers of first kind (A008275).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 8, 1;
%e 1, 10, 29, 24, 1;
%e ...
%p a := proc(n,k) option remember;
%p if k = 0 then 1
%p elif k < 0 then 0
%p elif k = n then (-1)^n
%p else a(n-1,k) - n*a(n-1,k-1) - (n-1)*a(n-2,k-2) fi end:
%p A046716 := (n,k) -> abs(a(n,k));
%p seq(seq(A046716(n,k),k=0..n),n=0..9); # _Peter Luschny_, Apr 05 2011
%t t[_, 0] = 1; t[n_, k_] := (-1)^k*Sum[(-1)^i*Binomial[n, i]*StirlingS1[i, n-k], {i, n-k, n}]; Table[t[n, k] // Abs, {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 10 2014 *)
%K nonn,tabl,easy
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jun 15 2004
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