A113132 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 = 5.

1, 1, 5, 50, 775, 16250, 426750, 13402500, 488566875, 20249281250, 939823431250, 48278138937500, 2719288331093750, 166652371531562500, 11040797013538437500, 786338134640203125000, 59916445436152444921875

Offset: 0

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

• nonn

Examples

			a(2) = 5.
a(3) = 2*5^2 = 50.
a(4) = 5*3*50 + 1*5*5 = 775.
a(5) = 5*4*775 + 1*5*50 + 2*50*5 = 16250.
a(6) = 5*5*16250 + 1*5*775 + 2*50*50 + 3*775*5 = 426750.
G.f.: A(x) = 1 + x + 5*x^2 + 50*x^3 + 775*x^4 + 16250*x^5 +...
= x/series_reversion(x + x^2 + 6*x^3 + 66*x^4 + 1056*x^5
+...).
		

Crossrefs

Cf. A008548, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113133(x=6), A113134(x=7), A113135(x=8).

Programs

Mathematica

x=5;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-2,5*j+1))))))[n+1]

PARI

a(n,x=5)=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} 5^k*A113129(n, k).

G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quintic factorials (A008548).

G.f. satisfies: A(x*G(x)) = G(x) = g.f. of quintic factorials (A008548).