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 1-5 of 5 results.

A005493 2-Bell numbers: a(n) = number of partitions of [n+1] with a distinguished block.

Original entry on oeis.org

1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 1192059223, 9097183602, 72384727657, 599211936355, 5150665398898, 45891416030315, 423145657921379, 4031845922290572, 39645290116637023, 401806863439720943, 4192631462935194064
Offset: 0

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Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.
a(n) = p(n+1) where p(x) is the unique degree n polynomial such that p(k) = A000110(k+1) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003
With offset 1, number of permutations beginning with 12 and avoiding 21-3.
Rows sums of Bell's triangle (A011971). - Jorge Coveiro, Dec 26 2004
Number of blocks in all set partitions of an (n+1)-set. Example: a(2)=10 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, with a total of 10 blocks. - Emeric Deutsch, Nov 13 2006
Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 2. - Olivier Gérard, Oct 29 2007
See page 29, Theorem 5.6 of my paper on the arXiv: These numbers are the dimensions of the homogeneous components of the operad called ComTrip associated with commutative triplicial algebras. (Triplicial algebras are related to even trees and also to L-algebras, see A006013.) - Philippe Leroux, Nov 17 2007
Number of set partitions of (n+2) elements where two specific elements are clustered separately. Example: a(1)=3 because 1/2/3, 1/23, 13/2 are the 3 set partitions with 1, 2 clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007
Equals A008277 * [1,2,3,...], i.e., the product of the Stirling number of the second kind triangle and the natural number vector. a(n+1) = row sums of triangle A137650. - Gary W. Adamson, Jan 31 2008
Prefaced with a "1" = row sums of triangle A152433. - Gary W. Adamson, Dec 04 2008
Equals row sums of triangle A159573. - Gary W. Adamson, Apr 16 2009
Number of embedded coalitions in an (n+1)-person game. - David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008
If prefixed with 0, gives first differences of Bell numbers A000110 (cf. A106436). - N. J. A. Sloane, Aug 29 2013
Sum_{n>=0} a(n)/n! = e^(e+1) = 41.19355567... (see A235214). Contrast with e^(e-1) = Sum_{n>=0} A000110(n)/n!. - Richard R. Forberg, Jan 05 2014

Examples

			For example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks and the distinguished block is parenthesized.

References

  • Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation. Unpublished as of 2017.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000110, A005494, A008277, A011971, A011968, A049020, A106436, A124323, A137650, A152433, A159573.
A row or column of the array A108087.
Row sums of triangle A143494. - Wolfdieter Lang, Sep 29 2011. And also of triangle A362924. - N. J. A. Sloane, Aug 10 2023

Programs

  • Maple
    with(combinat): seq(bell(n+2)-bell(n+1),n=0..22); # Emeric Deutsch, Nov 13 2006
seq(add(binomial(n, k)*(bell(n-k)), k=1..n), n=1..23); # Zerinvary Lajos, Dec 01 2006
A005493  := proc(n) local a,b,i;
a := [seq(3,i=1..n)]; b := [seq(2,i=1..n)];
2^n*exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=1,%),66)) end:
seq(A005493(n),n=0..22); # Peter Luschny, Mar 30 2011
BT := proc(n,k) option remember; if n = 0 and k = 0 then 1
elif k = n then BT(n-1,0) else BT(n,k+1)+BT(n-1,k) fi end:
A005493 := n -> add(BT(n,k),k=0..n):
seq(A005493(i),i=0..22); # Peter Luschny, Aug 04 2011
# For Maple code for r-Bell numbers, etc., see A232472. - N. J. A. Sloane, Nov 27 2013
  • Mathematica
    a=Exp[x]-1; Rest[CoefficientList[Series[a Exp[a],{x,0,20}],x] * Table[n!,{n,0,20}]]
a[ n_] := If[ n<0, 0, With[ {m = n+1}, m! SeriesCoefficient[ # Exp@# &[ Exp@x - 1], {x, 0, m}]]]; (* Michael Somos, Nov 16 2011 *)
Differences[BellB[Range[30]]] (* Harvey P. Dale, Oct 16 2014 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) + 2*x - 1), n))}; /* Michael Somos, Oct 09 2002 */
  • PARI
    {a(n) = if( n<0, 0, n+=2; subst( polinterpolate( Vec( serlaplace( exp( exp( x + O(x^n)) - 1) - 1))), x, n))}; /* Michael Somos, Oct 07 2003 */
  • Python
    # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A005493_list, blist, b = [], [1], 1
for _ in range(1001):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A005493_list.append(blist[-2])
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

Formula

a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.
E.g.f.: exp(exp(x) + 2*x - 1). First differences of Bell numbers (if offset 1). - Michael Somos, Oct 09 2002
G.f.: Sum_{k>=0} (x^k/Product_{l=1..k} (1-(l+1)x)). - Ralf Stephan, Apr 18 2004
a(n) = Sum_{i=0..n} 2^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007 [Written umbrally, a(n) = (B+2)^n. - N. J. A. Sloane, Feb 07 2009]
Representation as an infinite series: a(n-1) = Sum_{k>=2} (k^n*(k-1)/k!)/exp(1), n=1, 2, ... This is a Dobinski-type summation formula. - Karol A. Penson, Mar 14 2002
Row sums of A011971 (Aitken's array, also called Bell triangle). - Philippe Deléham, Nov 15 2003
a(n) = exp(-1)*Sum_{k>=0} ((k+2)^n)/k!. - Gerald McGarvey, Jun 03 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} (1+binomial(n, j))*a(j). - Jon Perry, Apr 25 2005
a(n) = A000296(n+3) - A000296(n+1). - Philippe Deléham, Jul 31 2005
a(n) = B(n+2) - B(n+1), where B(n) are Bell numbers (A000110). - Franklin T. Adams-Watters, Jul 13 2006
a(n) = A123158(n,2). - Philippe Deléham, Oct 06 2006
Binomial transform of Bell numbers 1, 2, 5, 15, 52, 203, 877, 4140, ... (see A000110).
Define f_1(x), f_2(x), ... such that f_1(x)=x*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Representation of numbers a(n), n=0,1..., as special values of hypergeometric function of type (n)F(n), in Maple notation: a(n)=exp(-1)*2^n*hypergeom([3,3...3],[2.2...2],1), n=0,1..., i.e., having n parameters all equal to 3 in the numerator, having n parameters all equal to 2 in the denominator and the value of the argument equal to 1. Examples: a(0)= 2^0*evalf(hypergeom([],[],1)/exp(1))=1 a(1)= 2^1*evalf(hypergeom([3],[2],1)/exp(1))=3 a(2)= 2^2*evalf(hypergeom([3,3],[2,2],1)/exp(1))=10 a(3)= 2^3*evalf(hypergeom([3,3,3],[2,2,2],1)/exp(1))=37 a(4)= 2^4*evalf(hypergeom([3,3,3,3],[2,2,2,2],1)/exp(1))=151 a(5)= 2^5*evalf(hypergeom([3,3,3,3,3],[2,2,2,2,2],1)/exp(1)) = 674. - Karol A. Penson, Sep 28 2007
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)charpoly(A,-2). - Milan Janjic, Jul 08 2010
a(n) = D^(n+1)(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A003128, A052852 and A009737. - Peter Bala, Nov 25 2011
From Sergei N. Gladkovskii, Oct 11 2012 to Jan 26 2014: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x*(k+3) - x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)-x) where U(k) = 1 - x - x*(k+1)/(1 - x/U(k+1)).
G.f.: G(0)/(1-x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+2*x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+3*x-1)/G(k+1) )).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-2*x-k*x)/(1-x/(x-1/G(k+1) )).
G.f.: -G(0)/x where G(k) = 1 - 1/(1-k*x-x)/(1-x/(x-1/G(k+1) )).
G.f.: 1 - 2/x + (1/x-1)*S where S = sum(k>=0, ( 1 + (1-x)/(1-x-x*k) )*(x/(1-x))^k / prod(i=0..k-1, (1-x-x*i)/(1-x) ) ).
G.f.: (1-x)/x/(G(0)-x) - 1/x where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ).
G.f.: (1/G(0) - 1)/x^3 where G(k) = 1 - x/(x - 1/(1 + 1/(x*k-1)/G(k+1) )).
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - x/(1 - x*(k+1)/Q(k+1)).
G.f.: G(0)/(1-3*x), where G(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1 - x*(k+3))*(1 - x*(k+4))/G(k+1) ). (End)
a(n) ~ exp(n/LambertW(n) + 3*LambertW(n)/2 - n - 1) * n^(n + 1/2) / LambertW(n)^(n+1). - Vaclav Kotesovec, Jun 09 2020
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) ~ n^2 * Bell(n) / LambertW(n)^2 * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021
a(n) = Sum_{k=0..n} 3^k*A124323(n, k). - Mélika Tebni, Jun 02 2022

Extensions

Definition revised by David Callan, Oct 11 2005

A362925 Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 15, 15, 13, 8, 1, 52, 52, 47, 35, 16, 1, 203, 203, 188, 153, 97, 32, 1, 877, 877, 825, 706, 515, 275, 64, 1, 4140, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1
Offset: 0

Views

Author

N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth

Keywords

Comments

A variant of A113547 and A362924. See those entries for further information.

Examples

			Triangle begins:
       1;
       1,      1;
       2,      2,      1;
       5,      5,      4,      1;
      15,     15,     13,      8,     1;
      52,     52,     47,     35,    16,     1;
     203,    203,    188,    153,    97,    32,     1;
     877,    877,    825,    706,   515,   275,    64,     1;
    4140,   4140,   3937,   3479,  2744,  1785,   793,   128,    1;
   21147,  21147,  20270,  18313, 15177, 11002,  6347,  2315,  256,   1;
  115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1;
  ...

Links

Crossrefs

Row sums are A000110(n+1).
Columns k=0+1,2-5 give A000110, A078468(n-2) (for n>=2), A383052(n-3) (for n>=3), A383053(n-4) (for n>=4), A383054(n-5) (for n>=5).
T(n+j,n) give (for j=0-2): A000012, A000079, A007689.
T(2n,n) gives A367820.
Cf. A040027, A113547, A362924.

Programs

  • Maple
    T:= (n, k)-> add(Stirling2(n-k, j)*(j+1)^k, j=0..n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 01 2023
  • Mathematica
    A362925[n_, m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
Table[A362925[n,m],{n,0,15},{m,0,n}] (* Paolo Xausa, Dec 04 2023 *)

Formula

Sum_{k=0..n} (k+1) * T(n,k) = A040027(n+1). - Alois P. Heinz, Dec 02 2023

A113547 Triangle read by rows: number of labeled partitions of n with maximin m.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 8, 13, 15, 15, 1, 16, 35, 47, 52, 52, 1, 32, 97, 153, 188, 203, 203, 1, 64, 275, 515, 706, 825, 877, 877, 1, 128, 793, 1785, 2744, 3479, 3937, 4140, 4140, 1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147, 21147, 1, 512, 6817, 23073, 45368, 68303, 88033, 102678, 111835, 115975, 115975
Offset: 1

Views

Author

Franklin T. Adams-Watters, Jan 13 2006

Keywords

Comments

The maximin of a partition is the maximum over all parts of the minimum label in each part. If the rows are reversed, the result is the number of partitions of n with minimax m.
The number of restricted growth functions of length n where the maximum appears first at position m. The RGF's are defined here as f(1)=1 and f(i) <=1+max_{1<=jR. J. Mathar, Mar 18 2016

Examples

			Maximin [123]=max(1)=1, maximin [12|3]=max(1,3)=3, maximin [13|2]=max(1,2)=2, maximin [1|23]=max(1,2)=2 and maximin [1|2|3]=max(1,2,3)=3, so for n=3 the multiset of maximins is {1,2,2,3,3}, making the 3rd line 1,2,2.
1;
1,  1;
1,  2,   2;
1,  4,   5,   5;
1,  8,  13,  15,  15;
1, 16,  35,  47,  52,  52;
1, 32,  97, 153, 188, 203, 203;
1, 64, 275, 515, 706, 825, 877, 877;

Links

Crossrefs

Cf. A008277, A000110, A271466.
A362924 and A362925 are other versions of this triangle. - N. J. A. Sloane, Aug 10 2023

Programs

  • Maple
    A113547 := proc(n,m)
    add(combinat[stirling2](m-1,k-1)*k^(n-m),k=1..m) ;
end proc:
seq(seq( A113547(n,m),m=1..n),n=1..10) ; # R. J. Mathar, Mar 13 2016
  • Mathematica
    T[n_, n_] := BellB[n - 1]; T[n_, n_ - 1] := BellB[n - 1]; T[n_, n_ - 2] := BellB[n - 1] - BellB[n - 3]; T[n_, m_] := Sum[StirlingS2[m - 1, k - 1]*k^(n - m), {k, 1, m}]; Table[T[n, m], {n, 1, 5}, {m, 1, n}] (* G. C. Greubel, May 06 2017 *)

Formula

T(n, m) = Sum_{k=1..m} S2(m-1, k-1)*k^(n-m), where S2 is the Stirling numbers of the second kind (A008277). T(n, n)=T(n, n-1)=B(n-1), where B is the Bell numbers (A000110). T(n, n-2)=B(n-1)-B(n-3).
Conjectures: T(n,3) = A007689(n-3). T(n,4) = 2^(n-4)+3^(n-3)+4^(n-4).- R. J. Mathar, Mar 13 2016

A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 1, 0, 1, 8, 5, -1, 0, 1, 16, 19, -3, 2, 0, 1, 32, 65, -1, 4, -6, 0, 1, 64, 211, 45, -10, -8, 24, 0, 1, 128, 665, 359, -116, 48, 20, -120, 0, 1, 256, 2059, 2037, -538, 340, -234, -52, 720, 0, 1, 512, 6305, 10079, -1316, 984, -1240, 1302, 72, -5040, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2025

Keywords

Examples

			Square array begins:
  1,  1,  1,    1,     1,     1,     1, ...
  1,  2,  4,    8,    16,    32,    64, ...
  0,  1,  5,   19,    65,   211,   665, ...
  0, -1, -3,   -1,    45,   359,  2037, ...
  0,  2,  4,  -10,  -116,  -538, -1316, ...
  0, -6, -8,   48,   340,   984, -1148, ...
  0, 24, 20, -234, -1240, -1866, 16400, ...

Links

Crossrefs

Columns k=0..6 give A019590(n+1), A302190 (for n > 0), A222627, A222636, A222748, A223023, A383050.
Main diagonal gives A383051.
Cf. A362924, A362925.

Programs

  • PARI
    a(n, k) = sum(j=0, n, (j+1)^k*stirling(n, j, 1));

Formula

A(n,k) = Sum_{j=0..n} (j+1)^k * Stirling1(n,j).
E.g.f. of column k: Sum_{j>=0} (j+1)^k * log(1+x)^j / j!.
E.g.f. of column k: (1+x) * Sum_{j=0..k} Stirling2(k+1,j+1) * log(1+x)^j.

A362926 Triangle read by rows: A113547 without its main diagonal.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 8, 13, 15, 1, 16, 35, 47, 52, 1, 32, 97, 153, 188, 203, 1, 64, 275, 515, 706, 825, 877, 1, 128, 793, 1785, 2744, 3479, 3937, 4140, 1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147, 1, 512, 6817, 23073, 45368, 68303, 88033, 102678, 111835, 115975, 1, 1024, 20195, 85475, 191866, 316305, 436297, 536882, 610989, 657423, 678570
Offset: 1

Views

Author

N. J. A. Sloane, Aug 11 2023, based on an email from Don Knuth

Keywords

Comments

A variant of A113547 and A362924. See those entries for further information.

Examples

			Triangle begins:
  [1],
  [1, 2],
  [1, 4, 5],
  [1, 8, 13, 15],
  [1, 16, 35, 47, 52],
  [1, 32, 97, 153, 188, 203],
  [1, 64, 275, 515, 706, 825, 877],
  [1, 128, 793, 1785, 2744, 3479, 3937, 4140],
  [1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147],
  ...

Crossrefs

Cf. A113547, A362924.

Programs

  • Mathematica
    A362926[n_,m_]:=Sum[StirlingS2[m-1,k-1]k^(n-m+1),{k,m}];
Table[A362926[n,m],{n,15},{m,n}] (* Paolo Xausa, Dec 02 2023 *)

Extensions

Name corrected by Paolo Xausa, Dec 02 2023
Showing 1-5 of 5 results.

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

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