cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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)).

Original entry on oeis.org

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

Views

Author

Emeric Deutsch, Feb 09 2011

Keywords

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).

Examples

			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; ...

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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

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

Original entry on oeis.org

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

Views

Author

Emeric Deutsch, Feb 09 2011

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

Cf. A184174, A184175, A184176, A000296.

Programs

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

Formula

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

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

Views

Author

Emeric Deutsch, Feb 09 2011

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

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

Formula

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

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