This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A366235 #16 Oct 08 2023 06:26:58
%S A366235 1,1,12,171,3644,104245,3718470,159587365,8014254120,461209324905,
%T A366235 29936339490050,2164061360402425,172443226346717100,
%U A366235 15018744392959920925,1419463584040707175950,144700081009666607896125,15826417814285141247938000,1848740412846656456007516625
%N A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(5*x*A(x)).
%C A366235 Related identities which hold formally for all Maclaurin series F(x):
%C A366235 (1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
%C A366235 (2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
%C A366235 (3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
%C A366235 (4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
%C A366235 (5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.
%C A366235 In general, if k > 0 and e.g.f. A(x) satisfies A(x) = 1 + x*A(x) * exp(k*x*A(x)), then a(n) ~ k^n * (1 + 2*LambertW(sqrt(k)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(k)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(k)/2)^(2*n + 3/2)). - _Vaclav Kotesovec_, Oct 06 2023
%F A366235 a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 5^k * (n-k)^k / k!.
%F A366235 Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!.
%F A366235 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A366235 (1) A(x) = 1 + x*A(x) * exp(5*x*A(x)).
%F A366235 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(5*x)) ).
%F A366235 (3) A( x/(1 + x*exp(5*x)) ) = 1 + x*exp(5*x).
%F A366235 (4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-5)*x*A(x)) for all fixed nonnegative m.
%F A366235 (4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-5)*x*A(x)).
%F A366235 (4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
%F A366235 (4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
%F A366235 (4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
%F A366235 (4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
%F A366235 (4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
%F A366235 a(n) ~ 5^n * (1 + 2*LambertW(sqrt(5)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(5)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(5)/2)^(2*n + 3/2)). - _Vaclav Kotesovec_, Oct 06 2023
%e A366235 E.g.f.: A(x) = 1 + x + 12*x^2/2! + 171*x^3/3! + 3644*x^4/4! + 104245*x^5/5! + 3718470*x^6/6! + 159587365*x^7/7! + 8014254120*x^8/8! + ...
%e A366235 where A(x) satisfies A(x) = 1 + x*A(x) * exp(5*x*A(x))
%e A366235 also
%e A366235 A(x) = 1 + 1^0*x*A(x)*exp(+4*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+3*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+2*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(+1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-0*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-1*x*A(x))/6! + ...
%e A366235 and
%e A366235 A(x) = 1 + 6*1*6^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 6*2*7^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 6*3*8^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 6*4*9^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 6*5*10^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
%t A366235 nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x] * E^(5*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* _Vaclav Kotesovec_, Oct 06 2023 *)
%o A366235 (PARI) /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
%o A366235 {a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!)}
%o A366235 for(n=0,20,print1(a(n),", "))
%o A366235 (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(5*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
%o A366235 for(n=0,20,print1(a(n),", "))
%Y A366235 Cf. A161633, A366232, A366233, A366234.
%Y A366235 Cf. A365775 (dual).
%K A366235 nonn
%O A366235 0,3
%A A366235 _Paul D. Hanna_, Oct 05 2023