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.

Previous Showing 11-17 of 17 results.

A125178 Triangle read by rows: T(n,0)=B(n) (the Bell numbers, A000110(n)), T(n,k)=0 for k < 0 or k > n, T(n,k) = T(n-1,k) + T(n-1,k-1) for n >= 1, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 15, 9, 7, 4, 1, 52, 24, 16, 11, 5, 1, 203, 76, 40, 27, 16, 6, 1, 877, 279, 116, 67, 43, 22, 7, 1, 4140, 1156, 395, 183, 110, 65, 29, 8, 1, 21147, 5296, 1551, 578, 293, 175, 94, 37, 9, 1, 115975, 26443, 6847, 2129, 871, 468, 269, 131, 46, 10, 1
Offset: 0

Views

Author

Gary W. Adamson, Nov 22 2006

Keywords

Comments

Row sums = 1, 2, 5, 13, 36, 109, 369, ...
Columns 0,1 and 2 yield A000110, A005001 and A029761, respectively.

Examples

			First few rows of the triangle:
    1;
    1,  1;
    2,  2,  1;
    5,  4,  3,  1;
   15,  9,  7,  4,  1;
   52, 24, 16, 11,  5, 1;
  203, 76, 40, 27, 16, 6, 1;
  ...
(4,3) = 16 = 7 + 9 = (3,3) + (3,2).

Crossrefs

Cf. A000110, A005001, A029761.

Programs

  • Maple
    with(combinat): T:=proc(n,k) if k=0 then bell(n) elif k<0 or k>n then 0 else T(n-1,k)+T(n-1,k-1) fi end: for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Extensions

Edited by N. J. A. Sloane, Nov 29 2006

A127568 Triangle T(n,k) = Bell(k) = A000110(k), 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 5, 15, 1, 1, 2, 5, 15, 52, 1, 1, 2, 5, 15, 52, 203
Offset: 0

Views

Author

Gary W. Adamson, Jan 19 2007

Keywords

Examples

			First few rows of the triangle are:
1;
1, 1;
1, 1, 2;
1, 1, 2, 5;
1, 1, 2, 5, 15;
1, 1, 2, 5, 15, 52;
...

Crossrefs

Cf. A005001 (row sums), A000110.

A137596 Triangle read by rows: T(n, k) = Sum_{i=0..n} Stirling2(i, k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 7, 1, 1, 5, 26, 32, 11, 1, 1, 6, 57, 122, 76, 16, 1, 1, 7, 120, 423, 426, 156, 22, 1, 1, 8, 247, 1389, 2127, 1206, 288, 29, 1, 1, 9, 502, 4414, 9897, 8157, 2934, 491, 37, 1
Offset: 0

Views

Author

Gary W. Adamson, Jan 29 2008

Keywords

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 2,   1;
  1, 3,   4,   1;
  1, 4,  11,   7,   1;
  1, 5,  26,  32,  11,   1;
  1, 6,  57, 122,  76,  16,  1;
  1, 7, 120, 423, 426, 156, 22, 1;
  ...

Crossrefs

Cf. A005001 (row sums), A048993.

Programs

  • Maple
    T := (n, k) -> add(Stirling2(i, k), i=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);  # Peter Luschny, Mar 07 2025

Formula

T = A000012 * A048993 as infinite lower triangular matrices.
T(n, k) = Sum_{i=0..n-k} Stirling2(i+k, k). - Igor Victorovich Statsenko, May 25 2024

Extensions

Offset set to 0 by Peter Luschny, May 25 2024

A171859 a(n) = Bell(n) - Sum_{j=0..n-1} Bell(j), where the Bell numbers are given in A000110.

Original entry on oeis.org

1, 0, 0, 1, 6, 28, 127, 598, 2984, 15851, 89532, 536152, 3392609, 22609852, 158220300, 1159380201, 8873605258, 70778190768, 587125257319, 5055713850058, 45114387675316, 416535887361323, 3973511993495144, 39112086371684844
Offset: 0

Views

Author

Emeric Deutsch, May 01 2010

Keywords

Comments

Number of partitions of the set {1,2,...,n} in which n is neither a singleton nor is in a block of consecutive integers. Example: a(4)=6 because we have 14-23, 13-24, 134-2, 124-3, 1-24-3, and 14-2-3. Note that if from the other partitions of {1,2,3,4}, namely 1234, 1-234, 12-34, 1-2-34, 123-4, 1-23-4, 12-3-4, 13-2-4, 1-2-3-4, we delete the blocks containing 4, then we are left with empty, 1, 12, 1-2, 123, 1-23, 12-3, 13-2, 1-2-3, i.e., all the partitions of the sets: empty, {1}, {1,2}, and {1,2,3}.
a(n) = A000110(n) - A005001(n).

Crossrefs

Cf. A000110, A005001.

Programs

  • Maple
    with(combinat): seq(bell(n)-add(bell(j), j = 0 .. n-1), n = 0 .. 23);

Formula

G.f.: G(0)*(1-x-x^2)/(1-x^2) + x/(1-x^2) where G(k) = 1 - x*(1-k*x)/(1 - x - x^2 - (1-2*x-x^2+2*x^3+x^4)/(1 - x - x^2 + (1-k*x)*(k*x+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 10 2013

A217143 Sum of squares of Bell numbers (A000110).

Original entry on oeis.org

1, 2, 6, 31, 256, 2960, 44169, 813298, 17952898, 465148507, 13915349132, 474372594032, 18228772272441, 782443669319410, 37224994809379094, 1949799331997896119, 111783178753323665728, 6978369826387194664144, 472207139326449254997425
Offset: 0

Views

Author

Emanuele Munarini, Sep 27 2012

Keywords

Crossrefs

Cf. A000110, A005001, A087650, A217144.
Partial sums of A001247.

Programs

  • Magma
    [&+[Bell(i)^2: i in [0..n]]: n in [0..20]]; // Bruno Berselli, Sep 27 2012
  • Mathematica
    Accumulate[BellB[Range[0, 20]]^2] (* Bruno Berselli, Sep 27 2012 *)
  • Maxima
    makelist(sum(belln(k)^2,k,0,n),n,0,30);
  • Python
    from itertools import accumulate, islice
def A217143_gen(): # generator of terms
    yield 1
    blist, b, c = (1,), 1, 1
    while True:
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield (c := c+b**2)
A217143_list = list(islice(A217143_gen(),20)) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = Sum_{k=0..n} Bell(k)^2.

A217144 Alternating sums of squares of Bell numbers (A000110).

Original entry on oeis.org

1, 0, 4, 21, 204, 2500, 38709, 730420, 16409180, 430786429, 13019414196, 447437830704, 17306961847705, 746907935199264, 35695643204860420, 1876878693983656605, 107956500727342113004, 6758630146906528885412, 458470139353155531447869
Offset: 0

Views

Author

Emanuele Munarini, Sep 27 2012

Keywords

Crossrefs

Cf. A000110, A005001, A087650, A217143.

Programs

  • Maxima
    makelist(sum((-1)^(n-k)*belln(k)^2,k,0,n),n,0,30);
  • Python
    from itertools import accumulate, islice
def A217144_gen(): # generator of terms
    yield 1
    blist, b, c, f = (1,), 1, 1, 1
    while True:
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield (f:=-f)*(c := c+f*b**2)
A217144_list = list(islice(A217144_gen(),20)) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k)^2.

A160181 Number of partitions of sets containing from 0 to n elements into blocks of at least 2 elements.

Original entry on oeis.org

1, 1, 2, 3, 7, 18, 59, 221, 936, 4361, 22083, 120336, 700653, 4333933, 28345090, 195233255, 1411303635, 10675375402, 84276173439, 692752181561, 5917018378496, 52416910416933, 480786834535247, 4559132648864256
Offset: 0

Views

Author

Anonymous, May 03 2009

Keywords

Comments

Partial sums of A000296.
a(n) is the total number of complete rhyme schemes for 0 to n lines; in other words, a(n) is the total number of rhyme schemes for 0 to n lines where each line rhymes with at least one other line.
If the restriction that the blocks of the partitions must have at least 2 elements is removed, then A005001 is obtained except for the first term of A005001.

Crossrefs

Cf. A000296, A005001, A000110.

Programs

  • Mathematica
    m=30; CoefficientList[Series[(1+x*Sum[x^k/Product[1-p*x, {p,0,k}], {k,0,m}])/(1-x^2), {x, 0,m}], x] (* Georg Fischer, Aug 28 2020 *)

Formula

G.f.: (G(0)-1)/(1-x) where G(k) = 1 + (1-x)/(1+x-x*k)/(1-x/(x+(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
G.f.: (1+x*sum{k>=0, x^k/prod[p=0..k, 1-p*x]})/(1-x^2). - Sergei N. Gladkovskii, Jan 25 2014

Extensions

a(22)-a(23) corrected by Georg Fischer, Aug 28 2020
Previous Showing 11-17 of 17 results.

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

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