A113133 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 6.

1, 1, 6, 72, 1332, 33264, 1040256, 38926656, 1692061488, 83688313536, 4638320578944, 284692939944192, 19169186341398912, 1404935464314299904, 111348880778746460160, 9489756817594314049536, 865470841829802331976448

Offset: 0

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

• nonn

Examples

			a(2) = 6.
a(3) = 2*6^2 = 72.
a(4) = 6*3*72 + 1*6*6 = 1332.
a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.
a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.
G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5
+...
= x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5
+...).
		

Crossrefs

Cf. A008542, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113134(x=7), A113135(x=8).

Programs

Mathematica

x=6;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 17}](Robert G. Wilson v)

PARI

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,6*j+1))))))[n+1]

PARI

a(n,x=6)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 6^k*A113129(n, k).

G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of sextuple factorial numbers (A008542).

G.f. satisfies: A(x*G(x)) = G(x) = g.f. of sextuple factorial numbers (A008542).