A113134 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 = 7.

1, 1, 7, 98, 2107, 61054, 2215094, 96203268, 4856212179, 279081882086, 17981777803682, 1283631249683804, 100557420457355358, 8577121056958121836, 791318123914138366924, 78521346319092948749576

Offset: 0

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

• nonn

Examples

			a(2) = 7.
a(3) = 2*7^2 = 98.
a(4) = 7*3*98 + 1*7*7 = 2107.
a(5) = 7*4*2107 + 1*7*98 + 2*98*7 = 61054.
a(6) = 7*5*61054 + 1*7*2107 + 2*98*98 + 3*2107*7 = 2215094.
G.f.: A(x) = 1 + x + 7*x^2 + 98*x^3 + 2107*x^4 + 61054*x^5
+...
= x/series_reversion(x + x^2 + 8*x^3 + 120*x^4 + 2640*x^5
+...).
		

Crossrefs

Cf. A045754, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113135(x=8).

Programs

Mathematica

x=7;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, 16}](Robert G. Wilson v)

PARI

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,7*j+1))))))[n+1]

PARI

a(n,x=7)=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} 7^k*A113129(n, k).

G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 7-fold factorials.

G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 7-fold factorials.