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A184174 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 2, i.e., blocks of the form (i,i+1) (0 <= k <= floor(n/2)).

%I A184174 #21 Sep 03 2017 18:42:02
%S A184174 1,1,1,1,3,2,10,4,1,35,14,3,139,54,9,1,611,224,38,4,2925,1027,171,16,
%T A184174 1,15128,5112,822,80,5,83903,27352,4279,415,25,1,495929,156392,23826,
%U A184174 2272,145,6,3108129,950285,141039,13252,855,36,1,20565721,6107540,883982,81692,5257,238,7
%N A184174 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 2, i.e., blocks of the form (i,i+1) (0 <= k <= floor(n/2)).
%C A184174 Row n contains 1 + floor(n/2) entries.
%C A184174 Sum of entries in row n = A000110(n) (the Bell numbers).
%C A184174 T(n,0) = A184175(n).
%C A184174 Sum_{k>=0} k*T(n,k) = A052889(n-1).
%H A184174 Alois P. Heinz, <a href="/A184174/b184174.txt">Rows n = 0..200, flattened</a>
%F A184174 T(n,k) = Sum_{j=k..floor(n/2)}(-1)^(k+j)*C(j,k)*C(n-j,j)*Bell(n-2j).
%F A184174 G.f.: A(x,y) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x + (1-y)*x^2). - _Paul D. Hanna_, Sep 03 2017
%e A184174 T(4,1)=4 because we have 12-3-4, 1-23-4, 1-2-34, 14-23. T(4,2)=1 because we have 12-34.
%e A184174 Triangle starts:
%e A184174 1;
%e A184174 1;
%e A184174 1, 1;
%e A184174 3, 2;
%e A184174 10, 4, 1;
%e A184174 35, 14, 3;
%e A184174 139, 54, 9, 1;
%e A184174 611, 224, 38, 4;
%e A184174 2925, 1027, 171, 16, 1;
%e A184174 15128, 5112, 822, 80, 5;
%e A184174 83903, 27352, 4279, 415, 25, 1;
%e A184174 495929, 156392, 23826, 2272, 145, 6;
%e A184174 3108129, 950285, 141039, 13252, 855, 36, 1; ...
%p A184174 with(combinat): q := 2: a := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*binomial(n+j-j*q, j)*bell(n-j*q), j = k .. floor(n/q)) end proc: for n from 0 to 13 do seq(a(n, k), k = 0 .. floor(n/q)) end do; # yields sequence in triangular form
%t A184174 T[n_, k_] := Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[n-j, j]*BellB[n-2j], {j, k, Floor[n/2]}]; Table[T[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* _Jean-François Alcover_, Feb 21 2017 *)
%o A184174 (PARI) {T(n,k) = my(A = sum(m=0,n, x^m/prod(k=0,m,1 - k*x + (1-y)*x^2 +x*O(x^n)))); polcoeff(polcoeff(A,n,x),k,y)}
%o A184174 for(n=0,12,for(k=0,n\2,print1(T(n,k),", "));print("")) \\ _Paul D. Hanna_, Sep 03 2017
%Y A184174 Cf. A000110, A184175, A052889, A124323, A184176, A184177.
%K A184174 nonn,tabf
%O A184174 0,5
%A A184174 _Emeric Deutsch_, Feb 09 2011