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.

A352682 Array read by ascending antidiagonals. A(n, k) = (n-1)*Gould(k-1) + Bell(k) for n >= 0 and k >= 1, A(n, 0) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 5, 6, 1, 4, 4, 8, 15, 21, 1, 5, 5, 11, 24, 52, 82, 1, 6, 6, 14, 33, 83, 203, 354, 1, 7, 7, 17, 42, 114, 324, 877, 1671, 1, 8, 8, 20, 51, 145, 445, 1400, 4140, 8536, 1, 9, 9, 23, 60, 176, 566, 1923, 6609, 21147, 46814
Offset: 0

Views

Author

Peter Luschny, Mar 28 2022

Keywords

Comments

The array defines a family of Bell-like sequences. The case n = 1 are the Bell numbers A000110, case n = 0 is A032347 and case n = 2 is A038561. The n-th sequence r(k) = T(n, k) is defined for k >= 0 by the recurrence r(k) = Sum_{j=0..k-1} binomial(k-1, j)*r(j) with r(0) = 1 and r(1) = n.

Examples

			Array starts:
n\k 0, 1,  2,  3,  4,   5,    6,    7,     8,      9, ...
---------------------------------------------------------
[0] 1, 0,  1,  2,  6,  21,   82,  354,  1671,   8536, ... A032347
[1] 1, 1,  2,  5, 15,  52,  203,  877,  4140,  21147, ... A000110
[2] 1, 2,  3,  8, 24,  83,  324, 1400,  6609,  33758, ... A038561
[3] 1, 3,  4, 11, 33, 114,  445, 1923,  9078,  46369, ... A038559
[4] 1, 4,  5, 14, 42, 145,  566, 2446, 11547,  58980, ... A352683
[5] 1, 5,  6, 17, 51, 176,  687, 2969, 14016,  71591, ...
[6] 1, 6,  7, 20, 60, 207,  808, 3492, 16485,  84202, ...
[7] 1, 7,  8, 23, 69, 238,  929, 4015, 18954,  96813, ...
[8] 1, 8,  9, 26, 78, 269, 1050, 4538, 21423, 109424, ...
[9] 1, 9, 10, 29, 87, 300, 1171, 5061, 23892, 122035, ...

Crossrefs

Rows: A032347, A000110 (Bell), A038561, A038559, A352683.
Diagonals: A352684 (main).
Cf. A040027 (Gould), A352686 (subtriangle).
Compare A352680 for a similar array based on the Catalan numbers.

Programs

  • Julia
    function BellRow(m, len)
    a = m; P = BigInt[1]; T = BigInt[1]
    for n in 1:len
        T = vcat(T, a)
        P = cumsum(vcat(a, P))
        a = P[end]
    end
T end
for n in 0:9 BellRow(n, 9) |> println end
  • Maple
    alias(PS = ListTools:-PartialSums):
BellRow := proc(n, len) local a, k, P, T;
a := n; P := [1]; T := [1];
for k from 1 to len-1 do
   T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
T end: seq(lprint(BellRow(n, 10)), n = 0..9);
  • Mathematica
    nmax = 10;
BellRow[n_, len_] := Module[{a, k, P, T}, a = n; P = {1}; T = {1};
   For[k = 1, k <= len - 1, k++,
      T = Append[T, a]; P = Accumulate[Join[{a}, P]]; a = P[[-1]]];
   T];
rows = Table[BellRow[n, nmax + 1], {n, 0, nmax}];
A[n_, k_] := rows[[n + 1, k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

Formula

Given a list T let PS(T) denote the list of partial sums of T. Given two list S and T let [S, T] denote the concatenation of the lists. Further let P[end] denote the last element of the list P. Row n of the array with length k can be computed by the following procedure:
A = [n], P = [1], R = [1];
Repeat k-1 times: R = [R, A], P = PS([A, P]), A = [P[end]];
Return R.

A038561 Left-hand border of triangle A046937.

Original entry on oeis.org

1, 2, 3, 8, 24, 83, 324, 1400, 6609, 33758, 185136, 1083233, 6726366, 44130128, 304741623, 2207682188, 16729947276, 132281116715, 1088831511000, 9311082630620, 82569723552561, 758057178490082, 7194283782101844, 70481938088367569
Offset: 0

Views

Author

Henry Gould, N. J. A. Sloane

Keywords

Comments

For n>1: a(n) is the number of entries in the last blocks of all set partitions of [n]. a(3) = 8 because the number of entries in the last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+1+1+2+1 = 8. - Alois P. Heinz, May 08 2017

References

  • H. W. Gould, A linear binomial recurrence and the Bell numbers and polynomials, preprint, 1998

Links

Crossrefs

A040027(n) + B(n), where B(n) = Bell numbers A000110.
Related to A000110, A040027, A038559, A038560.
Column k=1 of A286416 (for n>1).

Programs

  • Haskell
    a038561 = head . a046937_row  -- Reinhard Zumkeller, Jan 06 2014
  • Maple
    A038561List := proc(m) local A, P, n; A := [1,2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A038561List(24); # Peter Luschny, Mar 24 2022
  • Mathematica
    a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; a[n_] := a[n, 0]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 06 2013 *)

Formula

G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jun 30 2020

A038560 Binomial recurrence coefficients.

Original entry on oeis.org

2, 3, 5, 13, 39, 135, 527, 2277, 10749, 54905, 301111, 1761803
Offset: 0

Views

Author

Henry Gould

Keywords

Comments

Apparently defined by a(n) = A032347(n) + A038559(n). - R. J. Mathar, Nov 24 2013
Apparently the binomial transform of the sequence [2,5,13,39,...] is the same sequence without the "3". - R. J. Mathar, May 28 2019

References

  • H. W. Gould, A linear binomial recurrence and the Bell numbers and polynomials, preprint, 1998.

Links

Crossrefs

Related to A000110, A040027 and A038559.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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