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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)). 4
 1, 1, 1, 1, 3, 2, 10, 4, 1, 35, 14, 3, 139, 54, 9, 1, 611, 224, 38, 4, 2925, 1027, 171, 16, 1, 15128, 5112, 822, 80, 5, 83903, 27352, 4279, 415, 25, 1, 495929, 156392, 23826, 2272, 145, 6, 3108129, 950285, 141039, 13252, 855, 36, 1, 20565721, 6107540, 883982, 81692, 5257, 238, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row n contains 1 + floor(n/2) entries. Sum of entries in row n = A000110(n) (the Bell numbers). T(n,0) = A184175(n). Sum_{k>=0} k*T(n,k) = A052889(n-1). LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA T(n,k) = Sum_{j=k..floor(n/2)}(-1)^(k+j)*C(j,k)*C(n-j,j)*Bell(n-2j). 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 EXAMPLE 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. Triangle starts: 1; 1; 1, 1; 3, 2; 10, 4, 1; 35, 14, 3; 139, 54, 9, 1; 611, 224, 38, 4; 2925, 1027, 171, 16, 1; 15128, 5112, 822, 80, 5; 83903, 27352, 4279, 415, 25, 1; 495929, 156392, 23826, 2272, 145, 6; 3108129, 950285, 141039, 13252, 855, 36, 1; ... MAPLE 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 MATHEMATICA 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 *) PROG (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)} for(n=0, 12, for(k=0, n\2, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Sep 03 2017 CROSSREFS Cf. A000110, A184175, A052889, A124323, A184176, A184177. Sequence in context: A176743 A220466 A090780 * A277821 A318280 A057977 Adjacent sequences:  A184171 A184172 A184173 * A184175 A184176 A184177 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Feb 09 2011 STATUS approved

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Last modified May 26 07:44 EDT 2019. Contains 323579 sequences. (Running on oeis4.)