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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A128782 a(n) = n^2*4^n.

Original entry on oeis.org

0, 4, 64, 576, 4096, 25600, 147456, 802816, 4194304, 21233664, 104857600, 507510784, 2415919104, 11341398016, 52613349376, 241591910400, 1099511627776, 4964982194176, 22265110462464, 99230924406784, 439804651110400, 1939538511396864, 8514618045497344, 37225065669984256
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289, A007758, A086952.

Programs

Formula

From R. J. Mathar, Sep 20 2011: (Start)
G.f.: 4*x*(1 + 4*x)/(1 - 4*x)^3 .
a(n) = 4*A086952(n). (End)
E.g.f.: 4*exp(4*x)*x*(1 + 4*x). - Stefano Spezia, Oct 09 2022

A128789 n^3*2^n.

Original entry on oeis.org

0, 2, 32, 216, 1024, 4000, 13824, 43904, 131072, 373248, 1024000, 2725888, 7077888, 17997824, 44957696, 110592000, 268435456, 643956736, 1528823808, 3596091392, 8388608000, 19421724672, 44660948992, 102064193536, 231928233984
Offset: 0

Views

Author

Mohammad K. Azarian, Apr 07 2007

Keywords

Links

Crossrefs

Cf. A036289, A007758.

Programs

  • Magma
    [n^3*2^n: n in [0..30]]; // Vincenzo Librandi, Feb 07 2013
  • Mathematica
    CoefficientList[Series[2 x (1 + 8 x + 4 x^2)/(1 - 2 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 07 2013 *)
Table[n^3 2^n,{n,0,30}] (* or *) LinearRecurrence[{8,-24,32,-16},{0,2,32,216},30] (* Harvey P. Dale, Jun 14 2013 *)

Formula

G.f.: 2*x*(1 + 8*x + 4*x^2)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 07 2013
a(0)=0, a(1)=2, a(2)=32, a(3)=216, a(n)=8*a(n-1)-24*a(n-2)+ 32*a(n-3)- 16*a(n-4). - Harvey P. Dale, Jun 14 2013
E.g.f.: exp(2*x)*(2*x + 12*x^2 + 8*x^3). - Geoffrey Critzer, Aug 28 2013
Sum_{n>=1} 1/a(n) = (log(2))^3/6 - Pi^2*log(2)/12 + 7*Zeta(3)/8 = 0.53721319360804020094... . - Vaclav Kotesovec, Feb 15 2015

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020
Offset: 0

Views

Author

Zerinvary Lajos, Mar 25 2009

Keywords

Links

Crossrefs

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.
Cf. A038299, A126431.

Programs

Formula

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A228152 Triangle read by rows: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, n>=0, fibonacci(n+2)<=k<=2^n.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 24, 25, 30, 35, 40, 44, 49, 54, 59, 64, 50, 56, 62, 68, 73, 79, 85, 91, 97, 102, 107, 113, 119, 125, 131, 136, 142, 148, 154, 160, 96, 103, 110, 117, 123, 130, 137, 144, 151, 157, 163, 170, 177, 184, 191, 197, 204, 211, 218, 225, 231
Offset: 0

Views

Author

Herbert Eberle, Aug 13 2013

Keywords

Comments

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

Examples

			T(2,3) = 5 because in the (two) AVL trees of height 2 with 3 (leaf-) nodes one has depth 1 and two have depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so that the sum of depths is 5 for both trees.
Triangle begins:
  0
  . 2
  . . 5 8
  . . . . 12 16 20 24
  . . . .  .  .  . 25 30 35 40 44 49 54 59 64
  . . . .  .  .  .  .  .  .  .  . 50 56 62 68 73 79 85 91 97 102 ...
  . . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96 103 ...

References

  • D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Links

Crossrefs

Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Number of AVL trees read by rows gives: A143897.
Triangle read by columns gives: A228153.
The infimum of all external path lengths of binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Column maxima give: A228155(k).
Column heights give: A217710(k).
Number of AVL trees read by columns gives: A217298.

Programs

  • Maple
    with(combinat): F:=fibonacci:
T:= proc(n, k) option remember; `if`(n<1, 0, max(seq([k+T(n-1,t)+
      T(n-1,k-t), k+T(n-1,t) +T(n-2,k-t)][], t=F(n+1)..k-1)))
    end:
seq(seq(T(n, k), k=F(n+2)..2^n), n=0..7);  # Alois P. Heinz, Aug 14 2013
  • Mathematica
    maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[T[n, k], {n, 0, maxNods}, {k, 1, maxNods}] // Flatten // Select[#, IntegerQ]& (* Jean-François Alcover, Aug 14 2013, translated and adapted from Herbert Eberle's MuPAD program *)
  • MuPAD
    maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

A228153 Triangle read by columns: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, k>=1, A029837(k)<=n<A072649(k).

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 24, 25, 30, 35, 40, 44, 49, 50, 54, 56, 59, 62, 64, 68, 73, 79, 85, 91, 97, 96, 102, 103, 107, 110, 113, 117, 119, 123, 125, 130, 131, 137, 136, 144, 142, 151, 148, 157, 154, 163, 160, 170, 177, 184, 180, 191, 188, 197, 196, 204, 204
Offset: 1

Views

Author

Herbert Eberle, Aug 13 2013

Keywords

Comments

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

Examples

			In the (two) AVL trees of height 2 the 3 external nodes (leaves) have once depth 1 and twice depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so that the sum of depths is 5 for both trees.
Triangle begins:
  0
  . 2
  . . 5 8
  . . . . 12 16 20 24
  . . . .  .  .  . 25 30 35 40 44 49 54 59 64
  . . . .  .  .  .  .  .  .  .  . 50 56 62 68 73 79 85 91 97 102 ...
  . . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96 103 ...

References

  • D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Links

Crossrefs

Triangle read by rows gives: A228152.
Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Number of AVL trees read by rows gives: A143897.
The infimum of all external path lengths of binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Column maxima give: A228155(k).
Column heights give: A217710(k).
Number of AVL trees read by columns gives: A217298.

Programs

  • Mathematica
    maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[ Select[ Table[T[n, k], {n, A029837[k], A072649[k] - 1}], IntegerQ], {k, 1, maxNods}] // Flatten (* Jean-François Alcover, Aug 19 2013, translated and adapted from Herbert Eberle's MuPAD program *)
  • MuPAD
    maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

A228155 Maximal external path length of AVL trees with n (leaf-) nodes.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 25, 30, 35, 40, 44, 50, 56, 62, 68, 73, 79, 85, 91, 97, 103, 110, 117, 123, 130, 137, 144, 151, 157, 163, 170, 177, 184, 191, 197, 204, 211, 219, 227, 235, 243, 250, 257, 265, 273, 281, 289, 296, 304, 312, 320, 328, 335, 342, 349, 356
Offset: 1

Views

Author

Herbert Eberle, Aug 14 2013

Keywords

Comments

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

Examples

			The (two) AVL trees with 3 (leaf-) nodes have one with depth 1 and two with depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so a(3) = 5.

References

  • D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Links

Crossrefs

Column maxima of triangles A228152, A228153.
Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Column heights give: A217710(k).
Number of AVL trees read by rows gives: A143897.
The infimum of all external path lengths of all binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Number of AVL trees read by columns gives: A217298.

Programs

  • Mathematica
    maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[A228155[k], {k, 1, maxNods}] (* Jean-François Alcover, Aug 19 2013, translated and adapted from Herbert Eberle's MuPAD program *)
  • MuPAD
    maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

A286756 Irregular triangle T(n,k) for 0 <= k < 5n/2: T(n,k) = number of vertices of the cube-connected cycle graph of order n that are at a distance k from a designated vertex.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 6, 6, 3, 1, 1, 3, 5, 8, 11, 13, 13, 8, 2, 0, 1, 3, 6, 10, 16, 24, 31, 32, 23, 11, 3, 0, 1, 3, 6, 11, 18, 29, 43, 58, 72, 71, 47, 19, 5, 1, 0, 1, 3, 6, 12, 20, 34, 55, 83, 120, 154, 162, 131, 77, 29, 7, 2, 0
Offset: 1

Views

Author

Andrew Howroyd, May 13 2017

Keywords

Comments

The cube-connected cycle graph of order n is a vertex transitive graph with n*2^n vertices and degree 3.
The radius of the graph is floor(5n/2)-1 for n<=3 and floor(5n/2)-2 for n>3.

Examples

			Triangle starts:
1, 1
1, 2, 2, 2, 1
1, 3, 4, 6, 6, 3, 1
1, 3, 5, 8,  11, 13, 13, 8, 2, 0
1, 3, 6, 10, 16, 24, 31, 32, 23, 11, 3, 0
1, 3, 6, 11, 18, 29, 43, 58, 72, 71, 47, 19, 5, 1, 0
1, 3, 6, 12, 20, 34, 55, 83, 120, 154, 162, 131, 77, 29, 7, 2, 0
...
The order 3 graph has 24 vertices. For k=1 to 6 there are 3, 4, 6, 6, 3, 1 vertices at a distance k from any vertex in the graph.

Links

Crossrefs

Row sums are A036289.
Cf. A192191.

A340257 a(n) = 2^n * (1+n*(n+1)/2).

Original entry on oeis.org

1, 4, 16, 56, 176, 512, 1408, 3712, 9472, 23552, 57344, 137216, 323584, 753664, 1736704, 3964928, 8978432, 20185088, 45088768, 100139008, 221249536, 486539264, 1065353216, 2323644416, 5049942016, 10938744832, 23622320128, 50868518912, 109253230592, 234075717632
Offset: 0

Views

Author

Alois P. Heinz, Jan 02 2021

Keywords

Links

Crossrefs

Partial sums of A080929.
Cf. A000079, A000124, A001787, A001815, A036289, A087431, A340298.

Programs

  • Maple
    a:= n-> 2^n*(1+n*(n+1)/2):
seq(a(n), n=0..30);
  • Mathematica
    Table[2^n (1+(n(n+1))/2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{1,4,16},30] (* Harvey P. Dale, Jan 19 2023 *)

Formula

G.f.: (4*x^2-2*x+1)/(1-2*x)^3.
E.g.f.: exp(2*x)*(2*x^2+2*x+1).
a(n) = A000079(n) + A001815(n+1).
a(n) = A000079(n) * A000124(n).
a(n) = 2*a(n-1) + n*2^n = 2*a(n-1) + A036289(n), assuming a(-1) = 1/2.
a(n) = A340298(2^n).
a(n) = 2 * A087431(n) for n > 0.
a(n) = 4 * A007466(n) for n > 0.

A373339 Number of permutations in symmetric group S_n with an even number of cycles of length 2 or more.

Original entry on oeis.org

1, 1, 1, 1, 4, 36, 296, 2360, 19776, 180544, 1812352, 19953792, 239490560, 3113487872, 43589096448, 653837077504, 10461394714624, 177843713556480, 3201186851815424, 60822550202187776, 1216451004083601408, 25545471085844758528, 562000363888782868480
Offset: 0

Views

Author

Julian Hatfield Iacoponi, Jun 01 2024

Keywords

Examples

			a(1)=a(2)=a(3)=1 due to S_1,S_2,S_3 containing 1 permutation with an even number of non-fixed point cycles: the identity permutation, with 0 non-fixed point cycles.
a(4)=4 due to S_4 containing 4 permutations with an even number of non-fixed point cycles: the 3 (2,2)-cycles (12)(34),(13)(24),(14)(23); and the identity permutation (1)(2)(3)(4).

Crossrefs

Cf. A373340 (odd case), A000142, A001710, A036289.
Row sums of triangle A373417.

Programs

  • PARI
    a(n) = n!/2 - (n-2)*2^(n-2); \\ Michel Marcus, Jun 05 2024

Formula

a(n) = n!/2 - (n-2)*2^(n-2) = A001710(n) - A036289(n-2).
a(n) = A000142(n) - A373340(n).
E.g.f.: (1/(1 - x) + exp(2*x)*(1 - x))/2. - Stefano Spezia, Jun 05 2024

A373340 Number of permutations of symmetric group S_n with an odd number of cycles of length 2 or more.

Original entry on oeis.org

0, 0, 1, 5, 20, 84, 424, 2680, 20544, 182336, 1816448, 19963008, 239511040, 3113532928, 43589194752, 653837290496, 10461395173376, 177843714539520, 3201186853912576, 60822550206644224, 1216451004093038592, 25545471085864681472, 562000363888824811520
Offset: 0

Views

Author

Julian Hatfield Iacoponi, Jun 01 2024

Keywords

Examples

			a(0)=0 due to the sole permutation in S_0 being the empty permutation, with 0 non-fixed point cycles, not an odd number.
a(1)=0 due to the sole permutation in S_1 being the fixed point (1), with 0 non-fixed point cycles, not an odd number.
a(2)=1 due to 1 permutation in S_2 with an odd number of non-fixed point cycles: (12), with 1 non-fixed point cycle.
a(3)=5 due to 5 permutations in S_3 with an odd number of non-fixed point cycles: (12)(3),(13)(2),(23)(1),(123),(132), all with 1 non-fixed point cycle.

Crossrefs

Cf. A373339 (even case), A000142, A001710, A036289.
Row sums of triangle A373418.

Programs

  • PARI
    a(n) = n!/2 + (n-2)*2^(n-2); \\ Michel Marcus, Jun 05 2024

Formula

a(n) = n!/2 + (n-2)*2^(n-2) = A001710(n) + A036289(n-2).
a(n) = A000142(n) - A373339(n).
E.g.f.: (1/(1 - x) - exp(2*x)*(1 - x))/2. - Stefano Spezia, Jun 05 2024
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