A070779 Expansion of e.g.f.: (exp(x/(1-x))*(2-x)-1+x)/(1-x)^3.
1, 5, 28, 185, 1426, 12607, 125882, 1401409, 17209234, 231033431, 3365440882, 52855452817, 890097287834, 15996379554079, 305519496498106, 6178746162639617, 131885301216119842, 2962568890205560999, 69853182607494217154, 1724761580035969997521, 44501146220521229674282
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..442
- Mark Colarusso and Sam Evens, Orbits on a product of two flags and a line and the Bruhat Order, I, arXiv:2502.10912 [math.RT], 2025. See p. 16.
Programs
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Maple
f:= gfun:-rectoproc({(n + 3)*(n + 2)^2*a(n) - 3*(n + 3)^2*a(n + 1) + (3*n + 11)*a(n + 2) - a(n + 3)=0, a(0)=1,a(1)=5,a(2)=28},a(n),remember): map(f, [$0..30]); # Robert Israel, Jan 03 2019 # alternative A070779 := proc(n) n!*(n+1)^2*hypergeom([1,-n],[2,2],-1) ; simplify(%) ; end proc: # R. J. Mathar, Jul 16 2020 -
Mathematica
Table[(n + 1)! (LaguerreL[n + 1, -1] -1), {n, 0, 20}] (* Vincenzo Librandi, Jan 04 2019 *) With[{nn=20},CoefficientList[Series[(Exp[x/(1-x)](2-x)-1+x)/(1-x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 07 2020 *) -
Sage
@cached_function def a(n): if n < 3: return [1, 5, 28][n] return n*(n-1)^2*a(n-3)-3*n^2*a(n-2)+(3*n+2)*a(n-1) [a(n) for n in (0..20)] # Peter Luschny, Jan 04 2019
Formula
In Maple notation, a(n) = n! *(n+1)^2 *hypergeom([1, -n], [2, 2], -1).
a(n) = (n+1)!*(LaguerreL(n+1, -1)-1). - Vladeta Jovovic, Oct 24 2003
D-finite with recurrence a(n) = (3*n+2)*a(n-1) - 3*n^2*a(n-2) + n*(n-1)^2*a(n-3). - Robert Israel, Jan 03 2019
a(n) = Sum_{k=0..n} A355266(n+1, k+1). - Mélika Tebni, Jul 07 2022
Extensions
New description from Vladeta Jovovic, Apr 10 2003
Edited by Robert Israel, Jan 03 2019
Definition clarified by Harvey P. Dale, Sep 07 2020
Comments