A000387 -id:A000387 - cp's OEIS frontend

A211229 Matrix inverse of lower triangular array A211226.

1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, -2, 1, -1, 1, 0, 0, -1, 1, 2, -3, 3, 0, 0, -3, 1, -2, 2, -3, 3, 0, 0, -1, 1, 9, -8, 8, -12, 6, 0, 0, -4, 1, -9, 9, -8, 8, -6, 6, 0, 0, -1, 1, 44, -45, 45, -40, 20, -30, 10, 0, 0, -5, 1, -44, 44, -45, 45, -20, 20, -10, 10, 0, 0, -1, 1

Offset: 0

Peter Bala, Apr 05 2012

This triangle is related to the derangement numbers. The subtriangles (T(2*n,2*k))n,k>=0, -(T(2*n+1,2*k))n,k>=0, and (T(2*n+1,2*k+1))n,k>=0 are all equal to A008290, while the subtriangle (T(2*n,2*k+1))n,k>=0 equals -A180188 (with an extra initial row of zeros).

			Triangle begins:
   n\k |    0    1    2    3    4    5    6    7    8    9
  =====+==================================================
    0  |    1
    1  |   -1    1
    2  |    0   -1    1
    3  |    0    0   -1    1
    4  |    1    0    0   -2    1
    5  |   -1    1    0    0   -1    1
    6  |    2   -3    3    0    0   -3    1
    7  |   -2    2   -3    3    0    0   -1    1
    8  |    9   -8    8  -12    6    0    0   -4    1
    9  |   -9    9   -8    8   -6    6    0    0   -1    1
  ...

Cf. A000166, A000240, A000387, A000449, A000475, A008290, A180188, A211226.

b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
Table[(-1)^(n+k) (b[n]!/b[k]!) Sum[(-1)^i/i!, {i, 0, b[n-k]-h}], {n, 0, 31}, {k, 0, n}] //Flatten (* Manfred Boergens, Jan 10 2023 *)
(* Sum-free code *)
b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
T[n_, k_] = (-1)^(n+k) (b[n]!/b[k]!) If[n-k<2, 1, Round[(b[n-k]-h)!/E]/(b[n-k]-h)!];
Table[T[n, k], {n, 0, 31}, {k, 0, n}] // Flatten
(* Manfred Boergens, Jan 10 2023 *)

f(n) = (n\2)!; \\ A081123
T(n,k) = f(n)/(f(k)*f(n-k)); \\ A211226
tabl(nn) = my(m=matrix(nn, nn, n, k, if (n>=k, T(n-1,k-1), 0))); 1/m; \\ Michel Marcus, Jan 10 2023

T(2*n,2*k) = T(2*n+1,2*k+1) = -T(2*n+1,2*k) = binomial(n,k)*A000166(n-k) = (n!/k!)*Sum_{i = 0..n-k} (-1)^i/i!;
T(2*n,2*k+1) = -n*binomial(n-1,k)*A000166(n-k-1) = -(n!/k!)*Sum_{i = 0..n-k-1} (-1)^i/i!.
T(n,k) = T(n-k,0)*A211226(n,k).
Column entries:
T(2*n,0) = A000166(n), T(2*n,2) = A000240(n), T(2*n,4) = A000387(n), T(2*n,6) = A000449(n), T(2*n,8) = A000475(n).
From Manfred Boergens, Jan 10 2023: (Start)
With b(j) = floor(j/2); h = 1 for n even and k odd, h = 0 else:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*Sum_{i = 0..b(n-k)-h} (-1)^i/i!.
Sum-free formula:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!) for n-k < 2.
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*round((b(n-k)-h)!/exp(1))/(b(n-k)-h)!) otherwise. (End)

More terms from Manfred Boergens, Jan 10 2023

A228924 Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 1, 2, 1, 0, 0, 4, 8, 4, 10, 10, 6, 2, 0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1, 0, 0, 0, 6, 32, 44, 60, 118, 160, 208, 244, 244, 214, 174, 140, 104, 64, 30, 10, 2, 0, 0, 0, 1, 24, 83, 118, 206, 388, 565, 802, 1068, 1308, 1466, 1508, 1479, 1414, 1290, 1076, 806, 544, 333, 186, 96, 46, 19, 6, 1

Offset: 2

Geoffrey Critzer, Sep 08 2013

Row sums = A000166.
Sum_{k>=1} T(n,k)*k = A216239(n).
Sum_{even k} T(n,k) = A003221(n) and Sum_{odd k} T(n,k) = A000387(n).
It would be nice to have a closed formula for T(n,k). - Alois P. Heinz, Dec 31 2014

			Triangle T(n,k) begins:
  1;
  0, 2;
  0, 1, 4,  1,  2,  1;
  0, 0, 4,  8,  4, 10, 10,  6,  2;
  0, 0, 1, 12, 18, 16, 35, 44, 47, 40, 25, 14, 8, 4, 1;
  ...

Alois P. Heinz, Rows n = 2..23, flattened

Cf. A000166, A000387, A003221, A008302, A216239.

Map[Distribution[#,Range[1,Max[#]]]&,Table[Map[Inversions,Derangements[n]],{n,2,6}]]//Grid

A065088 a(n) = A000166(n)*binomial(n,2).

Original entry on oeis.org

0, 0, 1, 6, 54, 440, 3975, 38934, 415324, 4805856, 60073245, 807651350, 11630179506, 178681848696, 2918470195459, 50511984152070, 923647710209400, 17795612550034304, 360361154138194809, 7652375096699313126, 170052779926651402990, 3947014523560698353400

Offset: 0

N. J. A. Sloane, Nov 10 2001

nonn

Robert Israel, Table of n, a(n) for n = 0..446

Cf. A000387.

with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*binomial(n,2), n=0..21); # Zerinvary Lajos, Jun 11 2008

Table[Subfactorial[n]*Binomial[n, 2], {n, 0, 22}] (* Zerinvary Lajos, Jul 09 2009 *)

A009281 Expansion of e.g.f. exp(x)*cosh(log(1+x)).

Original entry on oeis.org

1, 1, 2, 1, 7, -19, 136, -923, 7421, -66743, 667486, -7342279, 88107427, -1145396459, 16035550532, -240533257859, 3848532125881, -65425046139823, 1177650830516986, -22375365779822543, 447507315596451071, -9397653627525472259, 206748379805560389952, -4755212735527888968619, 114125105652669335247157

Offset: 0

R. H. Hardin

Cf. A000387.

With[{nn=20},CoefficientList[Series[Exp[x]*Cosh[Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 02 2016 *)
CoefficientList[Series[(E^x*(1 + (1 + x)^2))/(2*(1 + x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 22 2015 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(x)*cosh(log(1+x)))) \\ Joerg Arndt, Jan 05 2024

a(n) ~ n! * (-1)^n / (2*exp(1)). - Vaclav Kotesovec, Jan 22 2015

a(n) = ((-1)^n * A000387(n)) + 1. - Christian Krause, Jan 05 2024

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Sep 02 2016

A182661 Expansion of x^3exp(-x)/(3(1-x)).

Original entry on oeis.org

2, 0, 20, 80, 630, 4928, 44520, 444960, 4894890, 58738240, 763597692, 10690366960, 160355505310, 2565688083840, 43616697426640, 785100553677888, 14916910519881810, 298338210397633920, 6265102418350314980, 137832253203706926480

Offset: 3

Geoffrey Critzer, Feb 01 2011

nonn

a(n) is the number of 3-cycles in all derangements of {1,2,...n}.

Cf. A000387.

egf:= x^3 * exp(-x)/(3*(1-x)):
a:= n-> n! * coeff (series (egf, x, n+1), x, n):
seq (a(n), n=3..25);

Table[Count[Flatten[Map[Length,Map[ToCycles,Derangements[n]],{2}]],3],{n,0,8}]
Range[0,20]! CoefficientList[Series[x^3/3 Exp[-x]/(1-x),{x,0,20}],x]

E.g.f.: x^3 * exp(-x)/(3*(1-x)).

In general, E.g.f. for the number of k cycles in the derangements of [n] is: x^k * exp(-x)/(k*(1-x)).

cp's OEIS Frontend

A211229 Matrix inverse of lower triangular array A211226.

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A228924 Irregular triangular array read by rows: T(n,k) is the number of derangement permutations of [n] that have exactly k inversions; n>=2, 1<=k<=binomial(n,2) for even n, 1<=k<=binomial(n,2)-1 for odd n.

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A065088 a(n) = A000166(n)*binomial(n,2).

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A009281 Expansion of e.g.f. exp(x)*cosh(log(1+x)).

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A182661 Expansion of x^3*exp(-x)/(3*(1-x)).

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A182661 Expansion of x^3exp(-x)/(3(1-x)).