A113131 - OEIS

%I #15 Dec 24 2024 22:14:22

%S 1,1,4,32,400,6784,144128,3658752,107686656,3599697920,134617038848,

%T 5567255822336,252278661832704,12431395516383232,661885541595873280,

%U 37869659304097218560,2317293119684500193280,151022143036329696952320

%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 = 4.

%F a(n+1) = Sum{k, 0<=k<=n} 4^k*A113129(n, k).

%F G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quartic factorials (A007696).

%F G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007696).

%e a(2) = 4.

%e a(3) = 2*4^2 = 32.

%e a(4) = 4*3*32 + 1*4*4 = 400.

%e a(5) = 4*4*400 + 1*4*32 + 2*32*4 = 6784.

%e a(6) = 4*5*6784 + 1*4*400 + 2*32*32 + 3*400*4 = 144128.

%e G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 400*x^4 + 6784*x^5 +...

%e = x/series_reversion(x + x^2 + 5*x^3 + 45*x^4 + 585*x^5 +...).

%t x=4;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, 18}](_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-2,4*j+1))))))[n+1]

%o (PARI) a(n,x=4)=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 Cf. A007696, A075834(x=1), A111088(x=2), A113130(x=3), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

%K nonn

%O 0,3

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