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%I #11 Jun 13 2017 22:42:27
%S 1,1,5,50,775,16250,426750,13402500,488566875,20249281250,
%T 939823431250,48278138937500,2719288331093750,166652371531562500,
%U 11040797013538437500,786338134640203125000,59916445436152444921875
%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 = 5.
%F a(n+1) = Sum{k, 0<=k<=n} 5^k*A113129(n, k).
%F G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quintic factorials (A008548).
%F G.f. satisfies: A(x*G(x)) = G(x) = g.f. of quintic factorials (A008548).
%e a(2) = 5.
%e a(3) = 2*5^2 = 50.
%e a(4) = 5*3*50 + 1*5*5 = 775.
%e a(5) = 5*4*775 + 1*5*50 + 2*50*5 = 16250.
%e a(6) = 5*5*16250 + 1*5*775 + 2*50*50 + 3*775*5 = 426750.
%e G.f.: A(x) = 1 + x + 5*x^2 + 50*x^3 + 775*x^4 + 16250*x^5 +...
%e = x/series_reversion(x + x^2 + 6*x^3 + 66*x^4 + 1056*x^5
%e +...).
%t 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_)
%o (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]
%o (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,
%o x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))
%Y Cf. A008548, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113133(x=6), A113134(x=7), A113135(x=8).
%K nonn
%O 0,3
%A _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005
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