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A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).

%I A033306 #28 Jul 08 2025 19:51:05
%S A033306 1,1,1,2,2,2,5,6,6,5,15,20,24,20,15,52,75,100,100,75,52,203,312,450,
%T A033306 500,450,312,203,877,1421,2184,2625,2625,2184,1421,877,4140,7016,
%U A033306 11368,14560,15750,14560,11368,7016,4140,21147,37260,63144,85260,98280,98280,85260,63144,37260,21147
%N A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).
%D A033306 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.
%F A033306 E.g.f.: exp(exp(x*y)+exp(x)-2).
%F A033306 Sum_{k=0..2n} (-1)^k * T(2n,k) = A000807(n). - _Alois P. Heinz_, Feb 13 2024
%e A033306    1;
%e A033306    1,  1;
%e A033306    2,  2,   2;
%e A033306    5,  6,   6,   5;
%e A033306   15, 20,  24,  20, 15;
%e A033306   52, 75, 100, 100, 75, 52;
%e A033306   ...
%p A033306 A033306 := proc(n,k)
%p A033306     if k < 0 or k > n then
%p A033306         0;
%p A033306     else
%p A033306         binomial(n,k)*combinat[bell](k)*combinat[bell](n-k) ;
%p A033306     end if;
%p A033306 end proc: # _R. J. Mathar_, Mar 21 2013
%p A033306 # second Maple program:
%p A033306 b:= proc(n) option remember; expand(`if`(n>0, add(
%p A033306      (x^j+1)*b(n-j)*binomial(n-1, j-1), j=1..n), 1))
%p A033306     end:
%p A033306 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p A033306 seq(T(n), n=0..10);  # _Alois P. Heinz_, Aug 30 2019
%t A033306 t[n_, k_] := Binomial[n, k] * BellB[k] * BellB[n-k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 14 2014 *)
%Y A033306 Cf. A000110, row sums give A001861.
%Y A033306 Columns include A000110 and A052889.
%Y A033306 Cf. A000807.
%K A033306 nonn,tabl,easy
%O A033306 0,4
%A A033306 _N. J. A. Sloane_
%E A033306 Edited by _Vladeta Jovovic_, Sep 17 2003