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 1-8 of 8 results.

A006505 Number of partitions of an n-set into boxes of size >2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=2 of A293024.
Cf. A000110, A000296, A057814, A057837, A059022.
Cf. A293038.

Programs

  • Maple
    Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # Zerinvary Lajos, Mar 13 2007
G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # Zerinvary Lajos, Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # N. J. A. Sloane, Jul 20 2011
  • Mathematica
    a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos, Jul 20 2011 *)
a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 09 2020 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos, Jul 20 2011 */

Formula

E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - R. J. Mathar, Nov 08 2008
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - Seiichi Manyama, Sep 22 2023

Extensions

More terms from Christian G. Bower, Nov 09 2000
Edited by N. J. A. Sloane, Jul 20 2011

A057814 Number of partitions of an n-set into blocks of size > 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 127, 463, 1255, 3004, 6722, 140570, 1039260, 5371627, 23202077, 90048525, 814737785, 7967774337, 62895570839, 417560407223, 2455461090505, 18440499041402, 179627278800426, 1770970802250146
Offset: 0

Views

Author

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Keywords

Links

Crossrefs

Column k=4 of A293024.
Row sums of A059024.
Cf. A000110, A000296, A006505, A057837.
Cf. A293040.

Programs

  • Maple
    G:={P=Set(Set(Atom,card>=5))}:combstruct[gfsolve](G,labeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..27); # Zerinvary Lajos, Dec 16 2007
  • Mathematica
    max = 27; CoefficientList[ Series[ Exp[ Exp[x] - Normal[ Series[ Exp[x], {x, 0, 4}]]], {x, 0, max}], x]*Range[0, max]!(* Jean-François Alcover, Apr 25 2012, from e.g.f. *)

Formula

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6-x^4/24).
a(0) = 1; a(n) = Sum_{k=5..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

A057837 Number of partitions of a set of n elements where the partitions are of size > 3.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 36, 127, 337, 793, 7525, 48764, 238954, 997790, 6401435, 49107697, 345482807, 2150694855, 14656830110, 116678887407, 978172378669, 7886661080873, 63905475745765, 553437891603452, 5122279358273976, 48331088541366296, 458771027309344261
Offset: 0

Views

Author

Steven C. Fairgrieve (fsteven(AT)math.wvu.edu), Nov 06 2000

Keywords

Links

Crossrefs

Column k=3 of A293024.
Row sums of A059023.
Cf. A000110, A000296, A006505, A057814.
Cf. A293039.

Programs

  • Maple
    G:={P=Set(Set(Atom,card>=4))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..26); # Zerinvary Lajos, Dec 16 2007
  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[Exp[x]-1-x-x^2/2-x^3/6],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 28 2012 *)

Formula

E.g.f.: exp(exp(x)-1-x-x^2/2-x^3/6).
a(0) = 1; a(n) = Sum_{k=4..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

Extensions

Corrected and extended by Christian G. Bower and James Sellers, Nov 09 2000

A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 6, 73, 1, 0, 0, 6, 36, 501, 1, 0, 0, 0, 24, 240, 4051, 1, 0, 0, 0, 24, 120, 1920, 37633, 1, 0, 0, 0, 0, 120, 1080, 17640, 394353, 1, 0, 0, 0, 0, 120, 720, 10080, 183120, 4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800, 2116800, 58941091
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Examples

			Square array begins:
    1,   1,   1,   1, ...
    1,   0,   0,   0, ...
    3,   2,   0,   0, ...
   13,   6,   6,   0, ...
   73,  36,  24,  24, ...
  501, 240, 120, 120, ...

Links

Crossrefs

Columns k=0..3 give A000262, A052845, A293049, A293050.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293024, A293119.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 29 2017
  • Mathematica
    A[0, ] = 1; A[n, k_] /; n <= k = 0; A[n_, k_] := A[n, k] = Sum[(i+1)! Binomial[n-1, i] A[n-1-i, k], {i, k, n-1}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 07 2020 *)
  • Ruby
    def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293053(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293053(20)

Formula

E.g.f. of column k: exp(x^(k+1)/(1-x)).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
A(n,k) = 2*(n-1) * A(n-1,k) - (n-1)*(n-2) * A(n-2,k) + (k+1)!*binomial(n-1,k) * A(n-1-k,k) - k*(k+1)!*binomial(n-1,k+1) * A(n-2-k,k) for n > k+1. - Seiichi Manyama, Mar 15 2023

A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, 0, -1, 2, -2, 1, 0, 0, 0, -1, 9, -9, 1, 0, 0, 0, -1, -1, 9, -9, 1, 0, 0, 0, 0, -1, 9, -50, 50, 1, 0, 0, 0, 0, -1, -1, 34, -267, 267, 1, 0, 0, 0, 0, 0, -1, -1, 90, -413, 413, 1, 0, 0, 0, 0, 0, -1, -1, 34, -71
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1,  0,  0,  0,  0, ...
    0, -1,  0,  0,  0, ...
    1, -1, -1,  0,  0, ...
    1,  2, -1, -1,  0, ...
   -2,  9, -1, -1, -1, ...

Links

Crossrefs

Columns k=0..4 give A000587, A293037, A293038, A293039, A293040.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A229223, A293024.

Formula

E.g.f. of column k: Product_{i>k} exp(-x^i/i!).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

A355247 Expansion of e.g.f. exp(2*(exp(x) - 1 + x)).

Original entry on oeis.org

1, 4, 18, 90, 494, 2946, 18926, 130066, 950654, 7353794, 59954638, 513333618, 4601380766, 43062556322, 419742815726, 4252083713874, 44680229906622, 486145710591874, 5468499473222670, 63503107472489266, 760281866742088670, 9373065303624742498, 118858898763010225198
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 25 2022

Keywords

Links

Crossrefs

Cf. A000110, A001861, A035009, A194689, A217924, A293024, A339014.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[2*Exp[x]-2+2*x], {x, 0, nmax}], x] * Range[0, nmax]!
Table[(BellB[n+2, 2] - BellB[n+1, 2])/4, {n, 0, 25}] (* Vaclav Kotesovec, Jul 21 2025 *)

Formula

a(n) ~ n^(n+2) * exp(n/LambertW(n/2) - n - 2) / (4 * sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n+2)).
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k+1) * Bell(n-k+1). - Ilya Gutkovskiy, Jun 26 2022

A293025 E.g.f.: exp(exp(x) - Sum_{i=0..5} x^i/i!).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 463, 1717, 4720, 11441, 25884, 56135, 2977313, 23524737, 125212889, 552517341, 2183244857, 8025931950, 124257251233, 1468856487536, 12433365625566, 85767520652726, 518324768774506, 2858925345803536, 26181976719735061
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Comments

a(n) is the number of set partitions of [n] into blocks of size > 5.

Links

Crossrefs

Column k=5 of A293024.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1), j=6..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 28 2017
  • Mathematica
    m = 31;
Exp[Exp[x] - Sum[x^i/i!, {i, 0, 5}]] + O[x]^m // CoefficientList[#, x]& // (# Range[0, m-1]!)& (* Jean-François Alcover, Mar 08 2021 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(exp(exp(x)-1-x-x^2/2-x^3/6-x^4/24-x^5/120)))

Formula

E.g.f.: Product_{i>5} exp(x^i/i!).

A336345 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(2 * (exp(x) - Sum_{j=0..k} x^j/j!)).

Original entry on oeis.org

1, 1, 2, 1, 0, 6, 1, 0, 2, 22, 1, 0, 0, 2, 94, 1, 0, 0, 2, 14, 454, 1, 0, 0, 0, 2, 42, 2430, 1, 0, 0, 0, 2, 2, 222, 14214, 1, 0, 0, 0, 0, 2, 42, 1066, 89918, 1, 0, 0, 0, 0, 2, 2, 142, 6078, 610182, 1, 0, 0, 0, 0, 0, 2, 2, 366, 36490, 4412798, 1, 0, 0, 0, 0, 0, 2, 2, 142, 3082, 238046, 33827974
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2020

Keywords

Examples

			Square array begins:
     1,   1,  1, 1, 1, 1, 1, ...
     2,   0,  0, 0, 0, 0, 0, ...
     6,   2,  0, 0, 0, 0, 0, ...
    22,   2,  2, 0, 0, 0, 0, ...
    94,  14,  2, 2, 0, 0, 0, ...
   454,  42,  2, 2, 2, 0, 0, ...
  2430, 222, 42, 2, 2, 2, 0, ...

Links

Crossrefs

Columns k=0..4 give A001861, A194689, A339014, A339017, A339027.
Main diagonal gives A000007.
Cf. A293024.

Programs

  • PARI
    {T(n, k) = n!*polcoef(prod(j=k+1, n, exp((x^j+x*O(x^n))/j!))^2, n)}
  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << 2 * (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
  ary
end
def A336345(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A336345(20)

Formula

E.g.f. of column k: (Product_{j>k} exp(x^j/j!))^2.
T(0,k) = 1, T(1,k) = T(2,k) = ... = T(k,k) = 0 and T(n,k) = 2 * Sum_{j=k..n-1} binomial(n-1,j)*T(n-1-j,k) for n > k.
Showing 1-8 of 8 results.

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

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