A393754 - OEIS

%I #7 Feb 28 2026 15:00:39

%S 1,1,9,217,10225,801681,94616761,15696482089,3483330195297,

%T 996272567289505,356793033395698921,156361716462978988281,

%U 82299934985279126998609,51225831690387051722811697,37215503820999440924493944985,31207282397036870862892419982921,29915379892820266366998272403668161

%N E.g.f.: exp( x * Sum_{n>=0} n! * 4^n * x^n ).

%H Paul D. Hanna, <a href="/A393754/b393754.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! along with F(x) = Sum_{n>=0} n! * x^n satisfy the following formulas.

%F (1) A(x) = exp( x*F(4*x) ).

%F (2) A(x) = exp(x + 4*x^2 * A'(x)/A(x)).

%F (3) A(x) = G(4*x)^(1/4) where G(x) is the g.f. of A158876.

%F (4) A'(x)/A(x) = Sum_{n>=0} (n+1)! * 4^n * x^n.

%F (5) [x^n] A(x)^(4*n) * (2 - F(4*x)) = 0 for n > 0.

%F (6) [x^n] A(x) * (4*n + 1 - F(4*x)) = 0 for n > 0.

%F a(n) = (n-1)! * Sum_{k=1..n} k! * 4^(k-1) * a(n-k) / (n-k)! for n > 0 with a(0) = 1.

%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 217*x^3/3! + 10225*x^4/4! + 801681*x^5/5! + 94616761*x^6/6! + 15696482089*x^7/7! + ...

%e where the logarithm of A(x) equals the integer series

%e log(A(x)) = x + 4*x^2 + 32*x^3 + 384*x^4 + 6144*x^5 + 122880*x^6 + ... + (n-1)!*4^(n-1)*x^n + ...

%e Let G(x) be the g.f. of A158876, then

%e A(x)^4 = G(4*x) = 1 + (4*x) + 3*(4*x)^2/2! + 19*(4*x)^3/3! + 217*(4*x)^4/4! + 4041*(4*x)^5/5! + 113611*(4*x)^6/6! + ... + A158876(n)*4^n*x^n/n! + ...

%o (PARI) {a(n) = n! * polcoef( exp(sum(k=1, n, (k-1)! * 4^(k-1) * x^k) +x*O(x^n)), n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k! * 4^(k-1) * a(n-k) / (n-k)!))}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A158876, A393752, A393753, A393755, A393756, A393757, A393758.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 27 2026