A113130 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 = 3.

1, 1, 3, 18, 171, 2214, 35910, 694980, 15567795, 395396478, 11218141170, 351527039676, 12056563337598, 449255267318844, 18074052522890604, 780881956274215944, 36062953309417344579, 1772992806860541951342

Offset: 0

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

• nonn

Examples

			a(2) = 3.
a(3) = 2*3^2 = 18.
a(4) = 3*3*18 + 1*3*3 = 171.
a(5) = 3*4*171 + 1*3*18 + 2*18*3 = 2214.
a(6) = 3*5*2214 + 1*3*171 + 2*18*18 + 3*171*3 = 35910.
G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 171*x^4 + 2214*x^5 +...
= x/series_reversion(x + x^2 + 4*x^3 + 28*x^4 + 280*x^5 +...).
		

Crossrefs

Cf. A007559, A075834(x=1), A111088(x=2), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

Programs

Mathematica

x=3;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, 18}](Robert G. Wilson v)

PARI

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,3*j+1))))))[n+1]

PARI

{a(n,x=3)=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} 3^k*A113129(n, k).

G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of triple factorials (A007559).

G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007559).