This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113135 #17 Dec 24 2024 22:13:54
%S A113135 1,1,8,128,3136,103424,4270080,211107840,12135936000,794618298368,
%T A113135 58355305676800,4749550536359936,424336070117163008,
%U A113135 41287521140173963264,4346005245162898325504,492102089936714946576384
%N A113135 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 = 8.
%F A113135 a(n+1) = Sum{k, 0<=k<=n} 8^k*A113129(n, k).
%F A113135 G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 8-fold factorials.
%F A113135 G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 8-fold factorials.
%e A113135 a(2) = 8.
%e A113135 a(3) = 2*8^2 = 128.
%e A113135 a(4) = 8*3*128 + 1*8*8 = 3136.
%e A113135 a(5) = 8*4*3136 + 1*8*128 + 2*128*8 = 103424.
%e A113135 a(6) = 8*5*103424 + 1*8*3136 + 2*128*128 + 3*3136*8 = 4270080
%e A113135 G.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3136*x^4 + 103424*x^5 +...
%e A113135 = x/series_reversion(x + x^2 + 9*x^3 + 153*x^4 + 3825*x^5 +...).
%t A113135 x=8;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_)
%o A113135 (PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,8*j+1))))))[n+1]
%o A113135 (PARI) a(n,x=8)=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))))))
%Y A113135 Cf. A045755, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7).
%K A113135 nonn
%O A113135 0,3
%A A113135 _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005