A396592 - OEIS

%I #12 May 30 2026 07:00:03

%S 1,2,22,479,15880,711977,40372828,2773990837,224184734080,

%T 20850884618561,2194931003624164,258095100967003445,

%U 33540797497557829360,4775042199876691978081,739225290389472515211076,123660840486107821484582093,22232061279653818569516521344,4275230842901062415074330238081

%N Expansion of e.g.f. F_2(x)/x and F_k(x) is the k-th iteration of x*G(x) with G(x) = exp(x*G(x)^4).

%F E.g.f.: (1/x) * Series_Reversion( H_2(x) ), where H_k(x) is the k-th iterate of U(x)*exp(-4*U(x)) and U(x) = -LambertW(-3*x)/3.

%F a(n) = Sum_{k=0..n} (k+1) * (4*k+1)^(k-1) * (4*n-3*k+1)^(n-k-1) * binomial(n,k).

%t a[n_]:=Sum[(k+1) * (4*k+1)^(k-1) * (4*n-3*k+1)^(n-k-1) * Binomial[n,k],{k,0,n}]; Array[a,18,0] (* _Stefano Spezia_, May 30 2026 *)

%o (PARI)

%o lista(nn, k=2, p=4, s=1, r=1) = {

%o   my(T=matrix(nn+1, nn+1, row, col, my(xr=row-1, xc=col-1); if(xc<xr, 0, (s*xr+r)*(p*xc-(p-s)*xr+r)^(xc-xr-1)*binomial(xc, xr))));

%o   my(TK=T^k);

%o   TK[1, ];

%o };

%Y Column k=2 of A396588.

%Y Cf. A101878, A396589, A396590.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 29 2026