OFFSET
1,2
COMMENTS
Related identity: Sum_{n>=0} (q^n + p)^n * r^n/n! = Sum_{n>=0} exp(p*q^n*r) * q^(n^2) * r^n/n!; here, q = A(x), p = -x, r = 5.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..250
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) 1 = Sum_{n>=0} 5^n * (A(x)^n - x)^n / n!.
(2) 1 = Sum_{n>=0} 5^n * exp(-5*x*A(x)^n) * A(x)^(n^2) / n!.
a(n) = Sum_{k=0..n-1} 5^k * A397350(n,k) for n >= 1.
EXAMPLE
E.g.f.: A(x) = x - 5*x^2/2! + 55*x^3/3! - 785*x^4/4! + 16875*x^5/5! - 469625*x^6/6! + 16497125*x^7/7! - 694705625*x^8/8! + 34067062375*x^9/9! + ...
where e.g.f. A(x) satisfies the following sums.
(1) 1 = 1 + 5*(A(x) - x) + 5^2*(A(x)^2 - x)^2/2! + 5^3*(A(x)^3 - x)^3/3! + 5^4*(A(x)^4 - x)^4/4! + 5^5*(A(x)^5 - x)^5/5! + ...
(2) 1 = exp(-5*x) + 5*exp(-5*x*A(x))*A(x) + 5^2*exp(-5*x*A(x)^2)*A(x)^4/2! + 5^3*exp(-5*x*A(x)^3)*A(x)^9/3! + 5^4*exp(-5*x*A(x)^4)*A(x)^16/4! + ...
SPECIFIC VALUES.
A(t) = 1/9 at t = 0.144546161490984356665014222597877054702420402968675...
A(1/7) = 0.110047195419307892700317017351678374658120904265652...
A(1/8) = 0.098606323736484495021085140401430801560381832219283...
A(1/9) = 0.089424211095130265725499051058520401406409278915909...
A(t) = -1/4 at t = -0.138647899807765217238750355200167615604807561897176...
A(-1/7) = -0.269710209144372206188129671761306633160414843306114...
A(-1/8) = -0.201817641643287589106868542107290660547043132798614...
A(-1/9) = -0.164891614131266852928898004505755918355430507365113...
PROG
(PARI) /* 1 = Sum_{n>=0} r^n * exp(-r*x*A(x)^n) * A(x)^(n^2) / n! */
{a(n, r=5) = my(A=[0, 1]); for(i=2, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=0, sqrtint(#A+1), r^m*exp(-r*Ser(A)^m*x +O(x^#A)) * Ser(A)^(m^2)/m! ), #A-1)/r; ); EGF=Ser(A); n!*A[n+1]}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(25)
(PARI) /* 1 = Sum_{n>=0} r^n * (A(x)^n - x)^n / n! */
{a(n, r=5) = my(A=[0, 1]); for(i=2, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A+1, r^m*(Ser(A)^m - x)^m/m! ), #A-1)/r ); EGF=Ser(A); n!*A[n+1]}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(25)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 22 2026
STATUS
approved
