A218672
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).
Original entry on oeis.org
1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047
Offset: 0
O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
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a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)
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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*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A218674
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(3*n)/n! * exp(-n*x*A(n*x)^3).
Original entry on oeis.org
1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672
Offset: 0
O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...
where
A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...
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^3,x,k*x)^k/k!*exp(-k*x*subst(A^3,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A218675
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(4*n)/n! * exp(-n*x*A(n*x)^4).
Original entry on oeis.org
1, 1, 5, 51, 817, 18562, 576687, 24203258, 1375038677, 106708683355, 11435867474152, 1708844338589752, 358640659116617571, 106261016900832212139, 44607231638918264608274, 26598477338494285370797703, 22569718290467849884279856477
Offset: 0
O.g.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 817*x^4 + 18562*x^5 + 576687*x^6 +...
where
A(x) = 1 + x*A(x)^4*exp(-x*A(x)^4) + 2^2*x^2*A(2*x)^8/2!*exp(-2*x*A(2*x)^4) + 3^3*x^3*A(3*x)^12/3!*exp(-3*x*A(3*x)^4) + 4^4*x^4*A(4*x)^16/4!*exp(-4*x*A(4*x)^4) + 5^5*x^5*A(5*x)^20/5!*exp(-5*x*A(5*x)^4) +...
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^4,x,k*x)^k/k!*exp(-k*x*subst(A^4,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
A218676
O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n*A(n*x)^(5*n)/n! * exp(-n*x*A(n*x)^5).
Original entry on oeis.org
1, 1, 6, 71, 1311, 34146, 1207717, 57298282, 3653975784, 316252925221, 37596625187796, 6206102367103899, 1434418185304457039, 466995106832397752352, 215051811411620578152401, 140491107719613466192347681, 130481943378389095603359529403
Offset: 0
O.g.f.: A(x) = 1 + x + 6*x^2 + 71*x^3 + 1311*x^4 + 34146*x^5 + 1207717*x^6 +...
where
A(x) = 1 + x*A(x)^5*exp(-x*A(x)^5) + 2^2*x^2*A(2*x)^10/2!*exp(-2*x*A(2*x)^5) + 3^3*x^3*A(3*x)^15/3!*exp(-3*x*A(3*x)^5) + 4^4*x^4*A(4*x)^20/4!*exp(-4*x*A(4*x)^5) + 5^5*x^5*A(5*x)^25/5!*exp(-5*x*A(5*x)^5) +...
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^5,x,k*x)^k/k!*exp(-k*x*subst(A^5,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
Showing 1-4 of 4 results.
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