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.

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A375870 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)^(3/2)) - 1) ).

Original entry on oeis.org

1, 2, 18, 310, 8038, 280264, 12313242, 653591922, 40704551630, 2910862397646, 235114931752898, 21172206066055312, 2103333121459719446, 228525476912967164714, 26957670075375556803178, 3431314158743477432894790, 468762478424957403561956702
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A349683, A375871, A375872.
Cf. A001861, A370056.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (3*n+2)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A349683.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * Stirling2(n,k).

A052896 E.g.f.: (exp(exp(x)-1)-1)^2.

Original entry on oeis.org

0, 0, 2, 12, 64, 350, 2024, 12460, 81638, 567888, 4180848, 32470834, 265219332, 2271692124, 20350705418, 190216812260, 1850993707960, 18714559108142, 196237054861920, 2130518566431620, 23912733627261670, 277078872201375976, 3310142647325149512
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.
a(n) is the number of ways to place n labeled balls into unlabeled (but two-colored) boxes so that at least one box is red and one box is blue. - Geoffrey Critzer, Oct 16 2011

Links

Crossrefs

Equals twice A000558.
Cf. A001861, A008277.

Programs

  • Maple
    spec := [S,{B=Set(Z,1 <= card),C=Set(B,1 <= card),S=Prod(C,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    a=Exp[Exp[x]-1]; Range[0,20]! CoefficientList[Series[(a-1)^2,{x,0,20}],x]

Formula

E.g.f.: exp(exp(x)-1)^2 - 2*exp(exp(x)-1) + 1.
For n >= 1: a(n) = Sum_{k=0...n} Stirling2(n,k)*(2^k-2) where Stirling2(n,k) is the number of set partitions of {1,2,...,n} into exactly k blocks (A008277).

Extensions

New name using e.g.f., Vaclav Kotesovec, Nov 20 2017

A129340 Triangular array read by rows: for n, k >= 1, a(n+1, 1) = 2*a(n, n); a(n+1, k+1) = a(n, k)+a(n+1, k).

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 22, 28, 36, 47, 94, 116, 144, 180, 227, 454, 548, 664, 808, 988, 1215, 2430, 2884, 3432, 4096, 4904, 5892, 7107, 14214, 16644, 19528, 22960, 27056, 31960, 37852, 44959, 89918, 104132, 120776, 140304, 163264, 190320, 222280
Offset: 1

Views

Author

Paul Curtz, May 28 2007

Keywords

Comments

Main diagonal is A035009. First column is A001861.

Crossrefs

Cf. A001861, A035009.

Formula

a(n, n) = A035009(n). For k < n, a(n, k) = 2*sum_{i = 1..k} binomial(k-1, i-1)*A035009(n-i).

Extensions

Edited and extended by David Wasserman, May 02 2008

A144061 Eigentriangle generated from A109128, row sums = expansion of {2(exp(x)-1)}.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 10, 6, 1, 7, 22, 42, 22, 1, 9, 38, 114, 198, 94, 1, 11, 58, 234, 638, 1034, 454, 1, 13, 82, 414, 1518, 3854, 5902, 2430, 1, 15, 110, 666, 3058, 10434, 24970, 36450, 14214, 1, 17, 142, 1002, 5522, 23594, 75818, 172530, 241638, 89918
Offset: 1

Views

Author

Gary W. Adamson & Roger L. Bagula, Sep 09 2008

Keywords

Comments

Row sums = A001861: (1, 2, 6, 22, 94, 454, 2430,...) = expansion of {2(exp(x)-1)}
Right border = A001861 shifted: (1, 1, 2, 6, 22, 94,...).
Sum of n-th row terms = rightmost term of next row.

Examples

			First few rows of the triangle =
1;
1, 1;
1, 3, 2;
1, 5, 10, 6;
1, 7, 22, 42, 22;
1, 9, 38, 114, 198, 94;
1, 11, 58, 234, 638, 1034, 454;
1, 13, 82, 414, 1518, 3854, 5902, 2430;
1, 15, 110, 666, 3058, 10434, 24970, 36450, 14214;
...
Example: row 3 = (1, 5, 10, 6) = termwise products of (1, 5, 5, 1) and (1, 1, 2, 6), where (1, 5, 5, 1) = row 3 of triangle A109128 and (1, 1, 2, 6) = the first 4 terms of A001861 shifted.

Crossrefs

A109128, Cf. A001861

Formula

T(n,k) = A109128(n,k)*A001861(k-1).
A109128 = (2*binomial(n,k) - 1): (1; 1,1; 1,3,1; 1,5,5,1;...).
A001861(k-1) = A001861 shifted one place, = (1, 1, 2, 6, 22, 94, 454,...).

A323632 Stirling transform of Jacobsthal numbers (A001045).

Original entry on oeis.org

0, 1, 2, 7, 31, 152, 813, 4741, 29956, 203305, 1470795, 11276718, 91221419, 775677177, 6910797962, 64326920851, 623981351195, 6293426736344, 65867162316433, 714062197266081, 8005397253530924, 92676194887133693, 1106385117766336919, 13603803900252612966, 172082332173918135687
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2019

Keywords

Links

Crossrefs

Cf. A000587, A001045, A001861, A263575, A263576.

Programs

  • Maple
    b:= proc(n, m) option remember;
     `if`(n=0, round(2^m/3), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Aug 06 2021
  • Mathematica
    nmax = 24; CoefficientList[Series[(Exp[2 (Exp[x] - 1)] - Exp[1 - Exp[x]])/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2^k - (-1)^k)/3, {k, 0, n}], {n, 0, 24}]
Table[(BellB[n, 2] - BellB[n, -1])/3, {n, 0, 24}]

Formula

E.g.f.: (exp(2*(exp(x) - 1)) - exp(1 - exp(x)))/3.
a(n) = Sum_{k=0..n} Stirling2(n,k)*A001045(k).
a(n) = (A001861(n) - A000587(n))/3.

A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 608, 3554, 22480, 152546, 1103200, 8456994, 68411632, 581745250, 5183126016, 48245682338, 467988498064, 4720072211938, 49400302118560, 535546012710434, 6004045485933104, 69507152958422370, 829789019700511040, 10202854323325253538, 129061753086335478736
Offset: 0

Views

Author

N. J. A. Sloane, Feb 16 2019

Keywords

Comments

Stirling transform of j-> ceiling(2^(j-2)). - Alois P. Heinz, Aug 25 2021

Links

Crossrefs

Cf. A000142, A001861, A035009, A347011.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
      ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Aug 25 2021
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0,
     Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

Formula

a(n) = -2^(n-1) + 2*Sum_{i = 0..n-1} binomial(n-1,i) * a(i) with a(0) = 1. [It follows from Kitaev's recurrence for C_n on p. 220 of his paper.] - Petros Hadjicostas, Oct 30 2019
From Alois P. Heinz, Aug 25 2021: (Start)
G.f.: Sum_{k>=0} ceiling(2^(k-2))*x^k / Product_{j=1..k} (1-j*x).
a(n) = Sum_{j=0..n} Stirling2(n,j)*ceiling(2^(j-2)). (End)

Extensions

More terms from Petros Hadjicostas, Oct 30 2019

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.

A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

Original entry on oeis.org

1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

Crossrefs

Cf. A000670, A001861, A052555, A080253, A083355, A216794, A346417, A346432.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001861(k) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * A000670(k).
a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021

A375867 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)^(1/2)) - 1) ).

Original entry on oeis.org

1, 2, 10, 82, 950, 14324, 266994, 5940218, 153797742, 4545958914, 151125136298, 5583189029004, 226989660492422, 10073099346726602, 484570780412539874, 25120235800280494530, 1396186059626363259038, 82828021612821756140124
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A001861, A030019, A065866.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (n+2)^(k-1)*stirling(n, k, 2));

Formula

a(n) = 2 * Sum_{k=0..n} (n+2)^(k-1) * Stirling2(n,k).

A375868 E.g.f. satisfies A(x) = exp( 2 * (exp(x*A(x)) - 1) ).

Original entry on oeis.org

1, 2, 14, 178, 3342, 83594, 2620998, 98968034, 4375295390, 221781470202, 12684194298998, 808136496137810, 56767509202678094, 4359070656483638762, 363283064756899367462, 32658326649544884611010, 3150270056733608259143422, 324571774149991316277596378
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A349598, A375869.
Cf. A001861, A370054.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (2*n+2)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A349598.
a(n) = 2 * Sum_{k=0..n} (2*n+2)^(k-1) * Stirling2(n,k).
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