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),", "))
A193363
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A((n+1)*x)^n/n! * exp(-(n+1)*x*A((n+1)*x)).
Original entry on oeis.org
1, 1, 4, 41, 871, 36137, 2885457, 443469511, 131707909982, 75945551138638, 85425571722359386, 188277619627892581987, 816318863956958720950775, 6986374103851011507327849798, 118360360643974268213872443877649, 3978536338453184605328853807076468581
Offset: 0
O.g.f.: A(x) = 1 + x + 4*x^2 + 41*x^3 + 871*x^4 + 36137*x^5 + 2885457*x^6 +...
where
A(x) = exp(-x*A(x)) + 2*x*A(2*x)*exp(-2*x*A(2*x)) + 3^2*x^2*A(3*x)^2/2!*exp(-3*x*A(3*x)) + 4^3*x^3*A(4*x)^3/3!*exp(-4*x*A(4*x)) + 5^4*x^4*A(5*x)^4/4!*exp(-5*x*A(5*x)) + 6^5*x^5*A(6*x)^5/5!*exp(-6*x*A(6*x)) +...
simplifies to a power series in x with integer coefficients.
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A[_] = 0; m = 16;
Do[A[x_] = Exp[-x A[x]] + Sum[(n+1)^n x^n A[(n+1)x]^n/n! Exp[-(n+1) x A[(n+1)x]], {n, 1, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)
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{a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, (k+1)*x)^k/k!*exp(-(k+1)*x*subst(A, x, (k+1)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A221410
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A(n*x)^n/n! * exp(-(n+1)*x*A(n*x)).
Original entry on oeis.org
1, 1, 3, 17, 160, 2209, 44081, 1247278, 50003383, 2843143785, 229717311597, 26423288336013, 4331881870569310, 1013060852125392519, 338180578288458076194, 161225876602752196310870, 109821236456762132613619651, 106923122485613725232770276036
Offset: 0
O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 160*x^4 + 2209*x^5 + 44081*x^6 +...
where
A(x) = exp(-x) + 2*x*A(x)*exp(-2*x*A(x)) + 3^2*x^2*A(2*x)^2/2!*exp(-3*x*A(2*x)) + 4^3*x^3*A(3*x)^3/3!*exp(-4*x*A(3*x)) + 5^4*x^4*A(4*x)^4/4!*exp(-5*x*A(4*x)) + 6^5*x^5*A(5*x)^5/5!*exp(-6*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+1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A221411
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+2)^n * x^n * A(n*x)^n/n! * exp(-(n+2)*x*A(n*x)).
Original entry on oeis.org
1, 1, 4, 27, 325, 5553, 140103, 4993445, 253780210, 18321882898, 1882677322704, 275715048156637, 57570654555092091, 17152947168669102772, 7295365645092117304955, 4430848642167010373612127, 3844378527942068170940925685, 4766454891141869269974497298382
Offset: 0
O.g.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 325*x^4 + 5553*x^5 + 140103*x^6 +...
where
A(x) = exp(-2*x) + 3*x*A(x)*exp(-3*x*A(x)) + 4^2*x^2*A(2*x)^2/2!*exp(-4*x*A(2*x)) + 5^3*x^3*A(3*x)^3/3!*exp(-5*x*A(3*x)) + 6^4*x^4*A(4*x)^4/4!*exp(-6*x*A(4*x)) + 7^5*x^5*A(5*x)^5/5!*exp(-7*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+2)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+2)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
A221412
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+3)^n * x^n * A(n*x)^n/n! * exp(-(n+3)*x*A(n*x)).
Original entry on oeis.org
1, 1, 5, 39, 576, 11693, 358649, 15411564, 951579001, 83392989241, 10419431480203, 1856210104355977, 471536928543684056, 170959559745467848287, 88469465053214549982042, 65371115770077488407503980, 68993903807593031325051425205, 104033290140443202579946504758992
Offset: 0
O.g.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 576*x^4 + 11693*x^5 + 358649*x^6 +...
where
A(x) = exp(-3*x) + 4*x*A(x)*exp(-4*x*A(x)) + 5^2*x^2*A(2*x)^2/2!*exp(-5*x*A(2*x)) + 6^3*x^3*A(3*x)^3/3!*exp(-6*x*A(3*x)) + 7^4*x^4*A(4*x)^4/4!*exp(-7*x*A(4*x)) + 8^5*x^5*A(5*x)^5/5!*exp(-8*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, 20, print1(a(n), ", "))
A221413
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+4)^n * x^n * A(n*x)^n/n! * exp(-(n+4)*x*A(n*x)).
Original entry on oeis.org
1, 1, 6, 53, 931, 21847, 791525, 39781921, 2896348222, 298603689072, 43979877929712, 9234821696038425, 2765498896234870783, 1182132922860352133076, 721128788569371093881079, 628104461090874688307332589, 781298529318782688558174387547
Offset: 0
O.g.f.: A(x) = 1 + x + 6*x^2 + 53*x^3 + 931*x^4 + 21847*x^5 + 791525*x^6 +...
where
A(x) = exp(-4*x) + 5*x*A(x)*exp(-5*x*A(x)) + 6^2*x^2*A(2*x)^2/2!*exp(-6*x*A(2*x)) + 7^3*x^3*A(3*x)^3/3!*exp(-7*x*A(3*x)) + 8^4*x^4*A(4*x)^4/4!*exp(-8*x*A(4*x)) + 9^5*x^5*A(5*x)^5/5!*exp(-9*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+4)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+4)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
Showing 1-6 of 6 results.
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