|
|
A193657
|
|
First difference of A002627.
|
|
4
|
|
|
1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
|
|
MAPLE
|
a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1, 1):
|
|
MATHEMATICA
|
q[n_, k_] := n + k + 1; (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}] (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2, {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Nov 20 2012 *)
|
|
PROG
|
(PARI) a(n) = { sum(k=0, n, if (k <= n-2, binomial(n, k)*(k+1)!, binomial(n, k)^2*k!)); } \\ Michel Marcus, Feb 07 2013
(Sage)
a = 2; b = 7; c = 31; n = 3
yield 1
while True:
yield a
n += 1
a, b, c = b, c, ((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|