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.

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A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 30, 21, 8, 1, 112, 88, 47, 12, 1, 463, 387, 253, 97, 17, 1, 2095, 1816, 1345, 675, 184, 23, 1, 10279, 9123, 7304, 4418, 1641, 324, 30, 1, 54267, 48971, 41193, 28396, 13276, 3645, 536, 38, 1, 306298, 279855, 243152, 183615, 102244, 36223, 7473, 842, 47, 1
Offset: 1

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Author

R. J. Mathar, Mar 13 2016

Keywords

Comments

The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i
T(n,p) is the number of elements in the p-th subset in all set partitions of [n]. - Joerg Arndt, Mar 14 2016

Examples

			The two restricted growth functions of length 2 are (1,1) and (1,2). The 1 appears 3 times and the 2 once, so T(2,1)=3 and T(2,2)=1.
1;
3,1;
9,5,1;
30,21,8,1;
112,88,47,12,1;
463,387,253,97,17,1;
2095,1816,1345,675,184,23,1;
10279,9123,7304,4418,1641,324,30,1;
54267,48971,41193,28396,13276,3645,536,38,1;
306298,279855,243152,183615,102244,36223,7473,842,47,1;
1838320,1695902,1506521,1211936,770989,334751,90223,14303,1267,57,1;
11677867,10856879,9799547,8237223,5795889,2965654,995191,207186,25820, 1839,68,1;
		

Crossrefs

Cf. A070071 (row sums).
T(2n+1,n+1) gives A270529.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->
          [p[1], p[2]+p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
        end:
    T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)[2]):
    seq(T(n), n=0..12);  # Alois P. Heinz, Mar 14 2016
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]] + p[[1]]*x^j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
    T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 0][[2]] ]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

Formula

T(n,n) = 1.
Conjecture: T(n,n-1) = 2+n*(n-1)/2 for n>1.
Conjecture: T(n+1,n-1) = 2+n*(n+1)*(3*n^2-5*n+26)/24 for n>1.
Sum_{k=1..n} k * T(n,k) = A346772(n). - Alois P. Heinz, Aug 03 2021

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262
Offset: 0

Author

Paul D. Hanna, Dec 31 2006, May 28 2007

Keywords

Comments

Limit of reversed rows equals A126348. Largest term in rows equal A126349.

Examples

			Number of terms in row n is: n*(n-1)/2 + 1.
Row functions B(n,q) begin:
  B(0,q) = 1;
  B(1,q) = 1;
  B(2,q) = 1 + q;
  B(3,q) = 1 + 2*q + q^2 + q^3;
  B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle begins:
  1;
  1;
  1, 1;
  1, 2, 1, 1;
  1, 3, 3, 4, 2, 1, 1;
  1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
  1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
  1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1;
  ...
		

Crossrefs

Row sums give A000110.
Cf. A126348, A126349; factorial variant: A126470.
Cf. A346772.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, x^t,
          add(b(n-1, max(m, j), t+j) , j=1..m+1))
        end:
    T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
    seq(T(n), n=0..8);  # Alois P. Heinz, Aug 02 2021
  • Mathematica
    B[0, ] = 1; B[n, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)
  • PARI
    {B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
    row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}
    
  • PARI
    /* Alternative formula for the n-th q-Bell number (row n): */ {B(n,q)=local(inf=100);round((0^n + sum(k=1, inf,((q^k-1)/(q-1))^n/prod(i=1,k,(q^i-1)/(q-1)))) / prod(k=1, inf,1 + (q-1)/q^k))}

Formula

G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021
Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025

Extensions

Keyword:tabl changed to tabf by R. J. Mathar, Oct 21 2010

A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 15, 25, 29, 31, 52, 89, 106, 115, 120, 203, 354, 431, 474, 499, 514, 877, 1551, 1923, 2141, 2273, 2355, 2407, 4140, 7403, 9318, 10489, 11224, 11695, 12002, 12205, 21147, 38154, 48635, 55286, 59595, 62434, 64331, 65614, 66491, 115975, 210803, 271617, 311469, 338019, 355951, 368205, 376665, 382559, 386699
Offset: 1

Author

Franklin T. Adams-Watters, Jun 06 2006, Jun 07 2006

Keywords

Examples

			The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9.
Table starts:
   1;
   2,  3;
   5,  8,   9;
  15, 25,  29,  31;
  52, 89, 106, 115, 120;
  ...
		

Crossrefs

Cf. A120058, A120095. First column is A000110.
Main diagonal is A087648(n-1).
Row sums give A346772.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, [1, 0],
          add((p-> [p[1], expand(p[2]*x+p[1]*j)])(
            b(n-1, max(m, j))), j=1..m+1))
        end:
    T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]):
    seq(T(n), n=1..10);  # Alois P. Heinz, Mar 24 2016
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
    T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]];
    Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 08 2016, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{i=1..k} A120058(n,i)*B(n-i+1), where B(n) are the Bell numbers, (A000110).
Showing 1-3 of 3 results.