A090012 Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.

3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465, 1432413063, 19743404469, 292164206259, 4619383947513, 77708277841575, 1385712098571957, 26108441941918851, 518231790473609481, 10808479322484810087

Offset: 1

Author

Jaap Spies, Dec 13 2003

Keywords

• nonn
• easy

References

• Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..447
• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

Crossrefs

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090013-A090016.

Programs

Maple

A090012 := proc(n,d) local r; if (n=1) then r := d+1 elif (n=2) then r := (d+1)^2 else r := (n+d-1)*A090012(n-1,d)+(n-2)*A090012(n-2,d) fi; RETURN(r); end: seq(A090012(n,2),n=1..20);

Mathematica

t={3,9};Do[AppendTo[t,(n+1)*t[[-1]]+(n-2)*t[[-2]]],{n,3,19}];t (* Indranil Ghosh, Feb 21 2017 *)
RecurrenceTable[{a[1]==3,a[2]==9,a[n]==(n+1)a[n-1]+(n-2)a[n-2]},a,{n,20}] (* Harvey P. Dale, Sep 21 2017 *)

Python

# Program to generate the b-file
print("1 3")
print("2 9")
a=3
b=9
c=(3+1)*b+(3-2)*a
for i in range(4, 40):
    print(str(i - 1)+" "+str(c))
    a=b
    b=c
    c=(i+1)*b+(i-2)*a # Indranil Ghosh, Feb 21 2017

Formula

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2), a(1)=3, a(2)=9.

a(n) = A000153(n-1) + A000153(n), a(1)=3.

G.f.: W(0)/x -1/x, where W(k) = 1 - x*(k+3)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013

a(n) ~ exp(-1) * n! * n^2 / 2. - Vaclav Kotesovec, Nov 30 2017

Extensions

Corrected by Jaap Spies, Jan 26 2004