A385546 E.g.f. A(x) satisfies A(x) = Sum_{k>=0} x^k/k! * A(2*k*x).
1, 1, 5, 73, 2841, 275841, 64191133, 34819498953, 43138054887665, 120180168811524673, 743714973197251893621, 10121932780068673030506249, 300492979488426431579059718281, 19324971141736799892467667226925889, 2676450415125002650227336306537165370253, 794234262095484293816451881720629809901365001
Offset: 0
Keywords
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n,k] * 2^k * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2*(i-j))^j*binomial(i, j)*v[j+1])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (2*(n-k))^k * binomial(n,k) * a(k).
a(n) ~ c * n! * (1 + sqrt(2))^n * 2^(n*(n-3)/2), where c = 0.890249679316305512447698257069341300111198385329042058845194612667048446402873... - Vaclav Kotesovec, Jul 03 2025