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A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

%I A144633 #28 Mar 27 2020 06:58:17
%S A144633 1,0,1,0,-1,1,0,2,-3,1,0,-5,11,-6,1,0,10,-45,35,-10,1,0,35,175,-210,
%T A144633 85,-15,1,0,-910,-315,1225,-700,175,-21,1,0,11935,-6265,-5670,5565,
%U A144633 -1890,322,-28,1,0,-134750,139755,-5005,-39270,19425,-4410,546,-36,1
%N A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.
%C A144633 Definition: take the triangle in A144385, write it as an (infinite) upper triangular square matrix, invert it and transpose it.
%C A144633 The Bell transform of A144636(n+1). Also the inverse Bell transform of the sequence "g(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%D A144633 J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.
%H A144633 Alois P. Heinz, <a href="/A144633/b144633.txt">Rows n = 0..140, flattened</a>
%H A144633 J. Y. Choi and J. D. H. Smith, <a href="http://dx.doi.org/10.1016/j.jcta.2005.10.001">The Tri-restricted Numbers and Powers of Permutation Representations</a>, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
%H A144633 J. Y. Choi and J. D. H. Smith, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00549-6">On the Unimodality and Combinatorics of the Bessel Numbers</a>, Discrete Math., 264 (2003), 45-53.
%H A144633 J. Y. Choi et al., <a href="http://dx.doi.org/10.1016/j.jcta.2005.10.001">Reciprocity for multirestricted Stirling numbers</a>, J. Combin. Theory 113 A (2006), 1050-1060.
%e A144633 Triangle begins:
%e A144633 1;
%e A144633 0,    1;
%e A144633 0,   -1,    1;
%e A144633 0,    2,   -3,    1;
%e A144633 0,   -5,   11,   -6,    1;
%e A144633 0,   10,  -45,   35,  -10,   1;
%e A144633 0,   35,  175, -210,   85, -15,   1;
%e A144633 0, -910, -315, 1225, -700, 175, -21,  1;
%p A144633 A:= proc(n,k) option remember; if n=k then 1 elif k<n or n<1 then 0 else A(n-1, k-1) +(k-1) *A(n-1, k-2) +(k-1) *(k-2) *A(n-1, k-3)/2 fi end:
%p A144633 M:= proc(n) option remember; Matrix(n+1, (i, j)-> A(i-1, j-1))^(-1) end:
%p A144633 T:= (n,k)-> M(n+1)[k+1, n+1]:
%p A144633 seq(seq(T(n,k), k=0..n), n=0..12); # _Alois P. Heinz_, Oct 23 2009
%t A144633 max = 10; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[_, _] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose; Table[A144633[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%o A144633 (Sage) # uses[bell_matrix from A264428]
%o A144633 bell_matrix(lambda n: A144636(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y A144633 For another version of this triangle see A144634.
%Y A144633 Columns give A144636-A144639.
%Y A144633 Cf. A144402.
%K A144633 sign,tabl
%O A144633 0,8
%A A144633 _N. J. A. Sloane_, Jan 21 2009
%E A144633 Corrected and extended by _Alois P. Heinz_, Oct 23 2009