A218681
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^2*x)^n/n! * exp(-n*x*A(n^2*x)).
Original entry on oeis.org
1, 1, 2, 17, 248, 8044, 499033, 62625238, 15947986557, 8220983161264, 8675909809528468, 18709697284980554577, 82551047593942653184220, 747564468621251440782891798, 13885138852461763218258064204207, 529723356811556257370919794910913765
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 248*x^4 + 8044*x^5 + 499033*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^2*x)^2/2!*exp(-2*x*A(2^2*x)) + 3^3*x^3*A(3^2*x)^3/3!*exp(-3*x*A(3^2*x)) + 4^4*x^4*A(4^2*x)^4/4!*exp(-4*x*A(4^2*x)) + 5^5*x^5*A(5^2*x)^5/5!*exp(-5*x*A(5^2*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k^2*x)^k/k!*exp(-k*x*subst(A,x,k^2*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A185029
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^4*x)^n/n! * exp(-n*x*A(n^4*x)).
Original entry on oeis.org
1, 1, 2, 65, 3524, 1364432, 1445333132, 7913299718555, 162327934705456532, 14083866155101076361024, 5251111824344114834186373747, 7956883819596423111541696080219295, 51760975171209084256721290749117849746987, 1424616119143714906580708999710589586791029920856
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 65*x^3 + 3524*x^4 + 1364432*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2^4*x)^2/2!*exp(-2*x*A(2^4*x)) + 3^3*x^3*A(3^4*x)^3/3!*exp(-3*x*A(3^4*x)) + 4^4*x^4*A(4^4*x)^4/4!*exp(-4*x*A(4^4*x)) + 5^5*x^5*A(5^4*x)^5/5!*exp(-5*x*A(5^4*x)) +...
simplifies to a power series in x with integer coefficients.
A219343
O.g.f. satisfies: A(x) = Sum_{n>=0} A(n*x)^n * (n^3*x)^n/n! * exp(-n^3*x*A(n*x)).
Original entry on oeis.org
1, 1, 32, 3183, 650929, 226009218, 119298668857, 89086101638412, 89480710389500666, 116491795770107486363, 191172400354899371561288, 387419202671209086703674709, 956322827450633453264262285623, 2859815748552720894795327258080881, 10430012061189048036456303441601971435
Offset: 0
O.g.f.: A(x) = 1 + x + 32*x^2 + 3183*x^3 + 650929*x^4 + 226009218*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(2*x)^2/2!*exp(-2^3*x*A(2*x)) + 3^9*x^3*A(3*x)^3/3!*exp(-3^3*x*A(3*x)) + 4^12*x^4*A(4*x)^4/4!*exp(-4^3*x*A(4*x)) + 5^15*x^5*A(5*x)^5/5!*exp(-5^3*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
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{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^(3*k)*x^k*subst(A,x,k*x)^k/k!*exp(-k^3*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
Showing 1-3 of 3 results.
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