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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.
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%I #11 Jun 13 2017 22:42:26

%S 1,1,6,72,1332,33264,1040256,38926656,1692061488,83688313536,

%T 4638320578944,284692939944192,19169186341398912,1404935464314299904,

%U 111348880778746460160,9489756817594314049536,865470841829802331976448

%N 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.

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

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

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

%e a(2) = 6.

%e a(3) = 2*6^2 = 72.

%e a(4) = 6*3*72 + 1*6*6 = 1332.

%e a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.

%e a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.

%e G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5

%e +...

%e = x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5

%e +...).

%t 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_)

%o (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]

%o (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,

%o x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

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

%K nonn

%O 0,3

%A _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005