A184174 -id:A184174 - cp's OEIS frontend

A184175 Number of set partitions of {1,2,...,n} having no blocks of the form {i, i+1}.

1, 1, 1, 3, 10, 35, 139, 611, 2925, 15128, 83903, 495929, 3108129, 20565721, 143134606, 1044489265, 7968879387, 63407648443, 525016067171, 4514661402304, 40245681692885, 371319303282381, 3540506731807277, 34840411462506887, 353394158240095874, 3690577066014598575

Offset: 0

Emeric Deutsch, Feb 09 2011

nonn

a(n) = A184174(n,0).

			a(3)=3 because we have 1-2-3, 13-2, and 123. a(4)=10 because among the 15 (=bell(4)) partitions of {1,2,3,4} only 12-34, 14-23, 12-3-4, 1-23-4, and 1-2-34, have adjacent blocks of size 2.
Contribution from _Paul D. Hanna_, Sep 03 2017: (Start)
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 35*x^5 + 139*x^6 + 611*x^7 + 2925*x^8 + 15128*x^9 + 83903*x^10 + 495929*x^11 + 3108129*x^12 +...
where
G.f.: A(x) = 1/(1+x^2) + x/((1+x^2)*(1-x+x^2)) + x^2/((1+x^2)*(1-x+x^2)*(1-2*x+x^2)) + x^3/((1+x^2)*(1-x+x^2)*(1-2*x+x^2)*(1-3*x+x^2)) +... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..550

Cf. A184174, A184175, A184176, A000296.

with(combinat): seq(add((-1)^j*binomial(n-j, j)*bell(n-2*j), j = 0 .. floor((1/2)*n)), n = 0 .. 25);

Table[Sum[(-1)^j*Binomial[n-j, j]*BellB[n-2*j], {j, 0, Floor[n/2]}], {n, 0, 25}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

{a(n) = my(A = sum(m=0,n, x^m/prod(k=0,m,1-k*x+x^2 +x*O(x^n)))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Sep 03 2017

a(n) = Sum((-1)^j*binomial(n-j,j)*bell(n-2j), j=0..floor(n/2)).

G.f.: Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x + 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).

A184177 Number of set partitions of {1,2,...,n} having no blocks of the form {i,i+1,i+2}.

Original entry on oeis.org

1, 1, 2, 4, 13, 46, 184, 805, 3840, 19775, 109180, 642382, 4007712, 26399764, 182939900, 1329327991, 10100670183, 80053631844, 660328296777, 5657449573120, 50255253534312, 462096537041953, 4391648990609487, 43079839534282880

Offset: 0

Emeric Deutsch, Feb 09 2011

nonn

a(n) = A184176(n,0).

			a(3) = 4 because we have 1-2-3, 1-23, 12-3, and 13-2.
a(4) = 13 because among the 15 (= Bell(4)) partitions of {1,2,3,4} only 123-4 and 1-234 have adjacent blocks of size 3.

Cf. A184176, A124323, A000296, A184174, A184175.

with(combinat): seq(add((-1)^j*binomial(n-2*j, j)*bell(n-3*j), j = 0 .. floor((1/3)*n)), n = 0 .. 23);

a(n) = Sum_{j=0..floor(n/3)} (-1)^j * binomial(n-2j,j) * Bell(n-3j,j).

cp's OEIS Frontend

A184175 Number of set partitions of {1,2,...,n} having no blocks of the form {i, i+1}.

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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)).

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Crossrefs

Programs

Maple

Mathematica

Formula

A184177 Number of set partitions of {1,2,...,n} having no blocks of the form {i,i+1,i+2}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula