Franel Number

Author

Eric W. Weisstein
October 4, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/FranelNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FranelNumber.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

Franel[n_]:=HypergeometricPFQ[{-n,-n,-n},{1,1},-1]

Enumeration

Franel/@Range[0,10]

Prime

(p=Flatten[Position[Franel/@Range[1000],_?PrimeQ]])//Timing

All even for n>0

Sums

(∑_(j=0)^n)((Binomial[n, j])^3)

V6

Sum[Binomial[n,j]^3,{j,0,n}]

V7

Sum[Binomial[n,j]^3,{j,0,n}]

(∑_(k=0)^n)((Binomial[n, k])^2) (Binomial[2 k, n])

V7

Sum[((Binomial[n,k])^2)Binomial[2k,n],{k,0,n}]

Generating Function

GeneratingFunction[HypergeometricPFQ[{-n,-n,-n},{1,1},-1],n,x]

Sum[HypergeometricPFQ[{-n,-n,-n},{1,1},-1](x^n),{n,0,∞}]

Integral

Table[(-1)^n Integrate[LaguerreL[n,x]^3 Exp[-x],{x,0,∞}],{n,0,6}]

V6

(-1)^n Integrate[LaguerreL[n,x]^3 Exp[-x],{x,0,∞}]

V7

(-1)^n Integrate[LaguerreL[n,x]^3 Exp[-x],{x,0,∞}]

Recurrence

a[0]:=1
a[1]:=2
a[n_]:=((8 (((-1+n))^2) a[-2+n]+(2-7 n+7 (n^2)) a[-1+n])/(n^2))

a/@Range[10]

V4.2

<<DiscreteMath`RSolve`

RSolve[{(n^2) a[n]==8 (((-1+n))^2) a[-2+n]+(2-7 n+7 (n^2)) a[-1+n],a[0]==1,a[1]==2},a[n],n]

V5

<<DiscreteMath`RSolve`

RSolve[{(n^2) a[n]==8 (((-1+n))^2) a[-2+n]+(2-7 n+7 (n^2)) a[-1+n],a[0]==1,a[1]==2},a[n],n]

V6

RSolve[{(n^2) a[n]==8 (((-1+n))^2) a[-2+n]+(2-7 n+7 (n^2)) a[-1+n],a[0]==1,a[1]==2},a[n],n]

V7

RSolve[{(n^2) a[n]==8 (((-1+n))^2) a[-2+n]+(2-7 n+7 (n^2)) a[-1+n],a[0]==1,a[1]==2},a[n],n]

Table[DifferenceRoot[Function[{𝓎,𝓃},{(-8-16 𝓃-8 (𝓃^2)) 𝓎[𝓃]+(-16-21 𝓃-7 (𝓃^2)) 𝓎[1+𝓃]+(4+4 𝓃+(𝓃^2)) 𝓎[2+𝓃]==0,𝓎[0]==1,𝓎[1]==2}]][n],{n,0,10}]

Table[HypergeometricPFQ[{-n,-n,-n},{1,1},-1],{n,0,10}]