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A261764 Triangle read by rows: T(n,k) is the number of nilpotent subpermutations on an n-set, each of nilpotency index less than or equal to k.

%I A261764 #35 Sep 29 2017 11:50:57
%S A261764 1,0,1,0,1,3,0,1,7,13,0,1,25,49,73,0,1,81,261,381,501,0,1,331,1531,
%T A261764 2611,3331,4051,0,1,1303,9073,19993,27553,32593,37633,0,1,5937,63393,
%U A261764 165873,253233,313713,354033,394353,0,1,26785,465769,1436473,2540233,3326473,3870793,4233673,4596553
%N A261764 Triangle read by rows: T(n,k) is the number of nilpotent subpermutations on an n-set, each of nilpotency index less than or equal to k.
%D A261764 A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.
%H A261764 Alois P. Heinz, <a href="/A261764/b261764.txt">Rows n = 0..140, flattened</a>
%F A261764 T(n, k) = T(n-1, k) + 2(n-1)T(n-2, k) + ... + k(n-1) ... (n-k+1)T(n-k, k), with T(n, 1) = 1 and T(n, n+r) = T(n, n) for every nonnegative integer r.
%F A261764 T(n,n) = A000262(n).
%F A261764 E.g.f. of column k: exp(x + x^2 + ... + x^k).
%F A261764 T(n,k) = Sum_{i=0..k} A157400(n,i).
%e A261764 T(3,2) = 7 because there are 7 nilpotent subpermutations on {1,2,3}, each of nilpotency index less than or equal to 2, namely: empty map, 1-->2, 1-->3, 2-->1, 2-->3, 3-->1, 3-->2.
%e A261764 Triangle starts:
%e A261764 1;
%e A261764 0, 1;
%e A261764 0, 1,   3;
%e A261764 0, 1,   7,   13;
%e A261764 0, 1,  25,   49,   73;
%e A261764 0, 1,  81,  261,  381,  501;
%e A261764 0, 1, 331, 1531, 2611, 3331, 4051;
%e A261764 ...
%p A261764 egf:= k-> exp(add(x^j, j=1..k)):
%p A261764 T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
%p A261764 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Oct 10 2015
%p A261764 # second Maple program:
%p A261764 T:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A261764       T(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n,k)))
%p A261764     end:
%p A261764 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Sep 29 2017
%t A261764 Table[n!*SeriesCoefficient[Exp[x*(x^k-1)/(x-1)], {x, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 18 2016 *)
%Y A261764 Cf. A000262, A157400, A261762, A261763, A261765, A261766, A261767.
%K A261764 nonn,tabl
%O A261764 0,6
%A A261764 _Samira Stitou_, Sep 21 2015
%E A261764 More terms from _Alois P. Heinz_, Oct 10 2015