A184177 -id:A184177 - cp's OEIS frontend

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)).

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

Offset: 0

Emeric Deutsch, Feb 09 2011

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).

			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; ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A184175, A052889, A124323, A184176, A184177.

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[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 *)

{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

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

A184176 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 3, i.e., blocks of the form (i,i+1,i+2) (0 <= k <= floor(n/3)).

Original entry on oeis.org

1, 1, 2, 4, 1, 13, 2, 46, 6, 184, 18, 1, 805, 69, 3, 3840, 288, 12, 19775, 1324, 47, 1, 109180, 6578, 213, 4, 642382, 35136, 1032, 20, 4007712, 200398, 5390, 96, 1, 26399764, 1214136, 30027, 505, 5, 182939900, 7778856, 177744, 2792, 30, 1329327991, 52501052, 1112969, 16362, 170, 1

Offset: 0

Emeric Deutsch, Feb 09 2011

Number of entries in row n is 1 + floor(n/3).
Sum of entries in row n = A000110(n) (the Bell numbers).
T(n,0) = A184177(n).
Sum_{k>=0}k*T(n,k) = A052889(n-2).

			T(4,1) = 2 because we have 123-4 and 1-234.
Triangle starts:
    1;
    1;
    2;
    4,  1;
   13,  2;
   46,  6;
  184, 18,  1;

Alois P. Heinz, Rows n = 0..250, flattened

Cf. A000110, A052889, A184174, A184175, A184177.

with(combinat): q := 3: 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 15 do seq(a(n, k), k = 0 .. floor(n/q)) end do; # yields sequence in triangular form

q = 3; a[n_, k_] := Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[n+j-j*q, j]* BellB[n-j*q], {j, k, Floor[n/q]}]; Table[a[n, k], {n, 0, 15}, {k, 0, Floor[n/q]}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

T(n,k) = Sum_{j=k..floor(n/3)}(-1)^(k+j) * C(j,k) * C(n-2j,j) * Bell(n-3j).

cp's OEIS Frontend

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)).

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