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%I #26 Feb 13 2024 11:30:11
%S 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 500,450,312,203,877,1421,2184,2625,2625,2184,1421,877,4140,7016,
%U 11368,14560,15750,14560,11368,7016,4140,21147,37260,63144,85260,98280,98280,85260,63144,37260,21147
%N Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.
%F E.g.f.: exp(exp(x*y)+exp(x)-2).
%F Sum_{k=0..2n} (-1)^k * T(2n,k) = A000807(n). - _Alois P. Heinz_, Feb 13 2024
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 5, 6, 6, 5;
%e 15, 20, 24, 20, 15;
%e 52, 75, 100, 100, 75, 52;
%e ...
%p A033306 := proc(n,k)
%p if k < 0 or k > n then
%p 0;
%p else
%p binomial(n,k)*combinat[bell](k)*combinat[bell](n-k) ;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 21 2013
%p # second Maple program:
%p b:= proc(n) option remember; expand(`if`(n>0, add(
%p (x^j+1)*b(n-j)*binomial(n-1, j-1), j=1..n), 1))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Aug 30 2019
%t 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 Cf. A000110, row sums give A001861.
%Y Columns include A000110 and A052889.
%Y Cf. A000807.
%K nonn,tabl,easy
%O 0,4
%A _N. J. A. Sloane_.
%E Edited by _Vladeta Jovovic_, Sep 17 2003
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