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 1-4 of 4 results.

A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 31, 10, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 344, 336, 140, 35, 21, 0, 1, 0, 1661, 1393, 616, 385, 56, 28, 0, 1, 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1, 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, May 12 2024

Keywords

Examples

			T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,4) = 0.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,    1;
  0,    31,    10,    10,    0,    1;
  0,    82,    70,    35,   15,    0,   1;
  0,   344,   336,   140,   35,   21,   0,   1;
  0,  1661,  1393,   616,  385,   56,  28,   0,  1;
  0,  7942,  6210,  4984, 1386,  504,  84,  36,  0, 1;
  0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
  ...

Links

Crossrefs

Columns k=0-2 give: A000007, A224219, A372764.
Row sums give A000110.
T(2n,n) gives A271425.
Cf. A372650, A372722.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

Formula

Sum_{k=0..n} k * T(n,k) = A372650(n).

A372721 Number of partitions of [n] having exactly one block of maximal size.

Original entry on oeis.org

0, 1, 1, 4, 11, 36, 132, 596, 2809, 14608, 79448, 461748, 2844052, 18559360, 127712483, 925057295, 7012810967, 55513992168, 457415487326, 3913510354554, 34702368052772, 318406785389976, 3018747693634775, 29537880351353635, 297953826680083794, 3095201088676962296
Offset: 0

Views

Author

Alois P. Heinz, May 11 2024

Keywords

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 36: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 12|3|4|5, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|23|4|5, 145|2|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Links

Crossrefs

Column k=1 of A372722.
Cf. A000110, A224219, A372802.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, 1, 0),
      add(binomial(n-1, j-1)*b(n-j, max(j, m),
     `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);
  • PARI
    A_x(N) = {my(x='x+O('x^N), f=x+sum(k=2,N, (x^k)/(k!)*exp(sum(j=1,k-1, (x^j)/(j!))))); concat([0],Vec(serlaplace(f)))}
A_x(30) \\ John Tyler Rascoe, Sep 09 2024

Formula

E.g.f: Sum_{k>0} ((x^k)/(k!) * exp(Sum_{j=1..k-1} (x^j)/(j!))). - John Tyler Rascoe, Sep 09 2024

A271715 Number of set partitions of [3n] with minimal block length multiplicity equal to n.

Original entry on oeis.org

1, 4, 55, 1540, 67375, 4239235, 383563180, 51925673800, 10652498631775, 3139051466175625, 1228555090548911125, 602267334323068414000, 357161594247065690582500, 250870551734754490461422500, 205672479804595549379158525000, 194557626586812183102927448930000
Offset: 0

Views

Author

Alois P. Heinz, Apr 12 2016

Keywords

Links

Crossrefs

Cf. A271424, A372722.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5,
      [1, 4, 55, 1540, 67375][n+1], ((2*(3*n-2))*
       (3*n-1)*(n^2-n-9)*a(n-1) -(3*(n-3))*(3*n-1)*
       (3*n-4)*(3*n-2)*(3*n-5)*a(n-2))/(4*n*(n-4)))
    end:
seq(a(n), n=0..20);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i&, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]];
a[n_] := If[n==0, 1, b[3n, 3n, n] - b[3n, 3n, n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz in A271424 *)

Formula

a(n) = A271424(3n,n).
Recursion: see Maple program.
For n>0, a(n) = (3^n + n!)*(3*n)! / (6^n * (n!)^2). - Vaclav Kotesovec, Apr 16 2016
a(n) = A372722(3n,n). - Alois P. Heinz, May 11 2024

A372649 Total sum over all partitions of [n] of the number of maximal blocks.

Original entry on oeis.org

0, 1, 3, 7, 21, 71, 293, 1268, 6107, 31123, 170745, 998966, 6212627, 40854360, 283290348, 2059884614, 15667307457, 124266461587, 1025342179759, 8784261413616, 78003593175261, 716854898767936, 6808817431686858, 66754426111124686, 674754718441688851
Offset: 0

Views

Author

Alois P. Heinz, May 08 2024

Keywords

Examples

			a(3) = 7 = 3 + 1 + 1 + 1 + 1: 1|2|3, 1|23, 12|3, 13|2, 123.
a(4) = 21 = 1+1+1+2+1+1+2+1+2+1+1+1+1+1+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

Links

Crossrefs

Cf. A000110, A097148, A372650, A372722.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, t,
      add(binomial(n-1, j-1)*b(n-j, max(j, m),
     `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..24);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
   Sum[Binomial[n - 1, j - 1]*b[n - j, Max[j, m],
   If[j > m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 10 2024, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A372722(n,k).
Showing 1-4 of 4 results.

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