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.

A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).

Original entry on oeis.org

1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478, 3319156078802044158, 40811417293301014150
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.
First column of the square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007
Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms, Apr 08 2007
Equals row sums of triangle A144061. - Gary W. Adamson, Sep 09 2008
Equals eigensequence of triangle A109128. - Gary W. Adamson, Apr 17 2009
Hankel transform is A108400. - Paul Barry, Apr 29 2009
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - Peter Bala, Mar 23 2013
The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M*A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - Tom Copeland, Apr 17 2014
Moments of the Poisson distribution with mean 2. - Vladimir Reshetnikov, May 17 2016
Exponential self-convolution of Bell numbers (A000110). - Vladimir Reshetnikov, Oct 06 2016

Examples

			a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are
01: [{1,2}] [ ];
02: [ ] [{1,2}];
03: [{1}] [{2}];
04: [{2}] [{1}];
05: [{1} {2}] [ ];
06: [ ] [{1} {2}].
- _Peter Bala_, Mar 23 2013

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For boxes of 1 color, see A000110, for 3 colors see A027710, for 4 colors see A078944, for 5 colors see A144180, for 6 colors see A144223, for 7 colors see A144263, for 8 colors see A221159.
First column of A078937.
Equals 2*A035009(n), n>0.
Row sums of A033306, A036073, A049020, and A144061.
Cf. A000110, A000587, A002871, A027710, A056857, A068199, A068200, A068201, A078937, A078938, A078944, A078945, A109128, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A213170, A221159, A221176.

Programs

  • Magma
    [&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2019
  • Maple
    A001861:=n->add(Stirling2(n,k)*2^k, k=0..n); seq(A001861(n), n=0..20); # Wesley Ivan Hurt, Apr 18 2014
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 04 2021
  • Mathematica
    Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* Geoffrey Critzer, Oct 06 2009 *)
mx = 16; p = 1; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[BellB[n, 2], {n, 0, 20}] (* Vaclav Kotesovec, Jan 06 2013 *)
  • PARI
    a(n)=if(n<0,0,n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)),n))
  • PARI
    {a(n)=polcoeff(sum(m=0, n, 2^m*x^m/prod(k=1,m,1-k*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */
  • PARI
    {a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019
  • Sage
    expnums(30, 2) # Zerinvary Lajos, Jun 26 2008

Formula

a(n) = Sum_{k=0..n} 2^k*Stirling2(n, k). - Emeric Deutsch, Oct 20 2001
a(n) = exp(-2)*Sum_{k>=1} 2^k*k^n/k!. - Benoit Cloitre, Sep 25 2003
G.f. satisfies 2*(x/(1-x))*A(x/(1-x)) = A(x) - 1; twice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - Gottfried Helms, Apr 08 2007
G.f.: 1/(1-2x-2x^2/(1-3x-4x^2/(1-4x-6x^2/(1-5x-8x^2/(1-6x-10x^2/(1-... (continued fraction). - Paul Barry, Apr 29 2009
O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Feb 15 2012
a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - Vaclav Kotesovec, Jan 06 2013
G.f.: (G(0) - 1)/(x-1)/2 where G(k) = 1 - 2/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
G.f.: ((1+x)/Q(0)-1)/(2*x), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-2*x-x*k)*(1-3*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) = Sum_{k=0..n} A033306(n,k) = Sum_{k=0..n} binomial(n,k)*Bell(k)*Bell(n-k), where Bell = A000110 (see Motzkin, p. 170). - Danny Rorabaugh, Oct 18 2015
a(0) = 1 and a(n) = 2 * Sum_{k=0..n-1} binomial(n-1,k)*a(k) for n > 0. - Seiichi Manyama, Sep 25 2017 [corrected by Ilya Gutkovskiy, Jul 12 2020]

A068199 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 114, 618, 3732, 24702, 177126, 1363740, 11195286, 97437138, 894857712, 8637708858, 87333790686, 922203924216, 10144109299146, 115972625504994, 1375221840671220, 16884112119546534, 214270296662325534, 2806600053170775372, 37892025089041181982
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

References

  • G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Crossrefs

Cf. A000110, A001861, this, A068200, A068201, ..., A000142.
Equals 2 * A027710(n).

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*g(n-k), k=1..n-1))*3)
    end:
a:= n-> `if`(n=0, 1, 2*g(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 09 2008
  • Mathematica
    a[n_] := 2*BellB[n-1, 3]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 28 2014 *)

Formula

E.g.f.: 1 + 2*exp(3exp(x)-3).

Extensions

More terms from Alois P. Heinz, Oct 09 2008

A068201 One of a family of sequences that interpolates between the Bell numbers and the factorials.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 4920, 37320, 309120, 2763720, 26440920, 268864320, 2889978120, 32689371720, 387638491920, 4803456571320, 62026296732120, 832595493624720, 11592967532632920, 167119530011233320, 2489936579950221120, 38283169922493447720
Offset: 0

Views

Author

N. J. A. Sloane, Mar 23 2002

Keywords

References

  • G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

Crossrefs

Cf. A000110, A001861, A068199, A068200, this, ..., A000142.

Programs

  • Maple
    g:= proc(n) option remember; `if` (n=0, 1, (1+add(binomial (n-1, k-1) * g(n-k), k=1..n-1)) * 5) end:
a:= n-> `if`(n<=2, (n+1)!, 24*g(n-3)): seq (a(n), n=0..25); # Sean A. Irvine, Feb 02 2024

Extensions

More terms from Sean A. Irvine, Feb 02 2024
Showing 1-3 of 3 results.

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

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