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 1-6 of 6 results.

A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570
Offset: 0

Views

Author

Alois P. Heinz, Oct 10 2008

Keywords

Comments

A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:
|1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|
|23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|
+--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+

Examples

			Square array A(n,k) begins:
   1,   1,  1,  1,  1,  1,  ...
   1,   1,  0,  0,  0,  0,  ...
   2,   3,  1,  0,  0,  0,  ...
   5,  10,  3,  1,  0,  0,  ...
  15,  41,  9,  4,  1,  0,  ...
  52, 196, 40, 10,  5,  1,  ...

Links

Crossrefs

Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
A(2n,n) gives A029651.
Cf.: A007318, A143398, A292948.

Programs

  • Maple
    exptr:= proc(p) local g; g:=
          proc(n) option remember; `if`(n=0, 1,
             add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))
        end: end:
A:= (n,k)-> exptr(i-> binomial(i, k))(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)
  • 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 << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A145460(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 A145460(20) # Seiichi Manyama, Sep 28 2017

Formula

A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017

A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 12 2008

Keywords

Examples

			T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
  1;
  0,   1;
  0,   3,  1;
  0,  10,  3,  1;
  0,  41,  9,  4,  1;
  0, 196, 40, 10,  5,  1;
  ...

Links

Crossrefs

Columns k=0-9 give: A000007, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
Main diagonal gives A000012.
Row sums give A143406.
T(2n,n) gives A029651.
Cf. A000142, A145460, A351703.

Programs

  • Maple
    u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    t[n_, n_] = 1; t[, 0] = 0; t[n, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)
  • PARI
    u(n,k) = if(k==0, 0, n\k);
T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

Formula

T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.

A346888 Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

Original entry on oeis.org

1, 0, 1, 3, 12, 70, 465, 3591, 31948, 319068, 3539385, 43205635, 575312826, 8298867798, 128921967265, 2145837600375, 38097353658120, 718657756980376, 14354000800751313, 302625047150614179, 6716038666999745710, 156498725047355717250, 3820426102008414736761
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Column k=2 of A351703.
Cf. A006153, A346889, A346890, A346893.
Cf. A000217, A133189, A308946, A353998.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x^2 Exp[x]/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ Michel Marcus, Aug 06 2021
  • PARI
    a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ Seiichi Manyama, May 13 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,2) * a(n-k).
a(n) ~ n! / ((1 + LambertW(1/sqrt(2))) * 2^(n+1) * LambertW(1/sqrt(2))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k)/(2^k * (n-2*k)!). - Seiichi Manyama, May 13 2022

A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 3, -20, -135, -189, 3598, 33300, 39105, -2164085, -23831214, -5268042, 3038813869, 36984819795, -59749871880, -8207734934984, -105142191601887, 482549202944307, 37754304692254030, 489494512692093090, -4466445363328684659, -271973408844483808517
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Links

Crossrefs

Cf. A000217, A009306, A133189, A300455, A346751, A346752, A346753.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 + x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 2] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 0; a(n) = binomial(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

Crossrefs

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - x^2 Exp[x]/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 2] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A292909 E.g.f.: exp(1/2! * x^2 * exp(-x)).

Original entry on oeis.org

1, 0, 1, -3, 9, -40, 210, -1176, 7273, -49932, 372060, -2971540, 25359411, -230364498, 2215550428, -22460391240, 239236043985, -2669869110856, 31134833803728, -378485082644400, 4786085290280275, -62838103267148790, 855122923978737876
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Links

Crossrefs

Column k=2 of A292948.
Cf. A133189, A292907.

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(1/2!*x^2*exp(-x))))

Formula

a(n) = (-1)^n * A133189(n).
Showing 1-6 of 6 results.

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