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 A113133 #15 Dec 24 2024 22:13:50
%S A113133 1,1,6,72,1332,33264,1040256,38926656,1692061488,83688313536,
%T A113133 4638320578944,284692939944192,19169186341398912,1404935464314299904,
%U A113133 111348880778746460160,9489756817594314049536,865470841829802331976448
%N A113133 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 A113133 a(n+1) = Sum{k, 0<=k<=n} 6^k*A113129(n, k).
%F A113133 G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of sextuple factorial numbers (A008542).
%F A113133 G.f. satisfies: A(x*G(x)) = G(x) = g.f. of sextuple factorial numbers (A008542).
%e A113133 a(2) = 6.
%e A113133 a(3) = 2*6^2 = 72.
%e A113133 a(4) = 6*3*72 + 1*6*6 = 1332.
%e A113133 a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.
%e A113133 a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.
%e A113133 G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5
%e A113133 +...
%e A113133 = x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5
%e A113133 +...).
%t A113133 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 A113133 (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 A113133 (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,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))
%Y A113133 Cf. A008542, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113134(x=7), A113135(x=8).
%K A113133 nonn
%O A113133 0,3
%A A113133 _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005