A193657 First difference of A002627.
1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
Programs
-
Maple
a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1): seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014
-
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 *)
-
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
def A193657(): 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 a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014
Formula
E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
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
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
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.
Second column of A176305. (End)
Extensions
Simpler definition by Peter Luschny, May 30 2014
Comments