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A361311
G.f. A(x) satisfies A(x) = Series_Reversion(x - x^5*A'(x)).
7
1, 1, 10, 195, 5520, 201255, 8881551, 457227585, 26805712005, 1759840463070, 127784731466660, 10164274303786460, 878859905526721250, 82080454974318915935, 8235485665033295289810, 883569144560890419421630, 100952601749463417250801935, 12239031817482031919864850550
OFFSET
1,3
LINKS
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^(4*n-3) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^5*A'(x)).
(2) A(x) = x + A(x)^5 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A'(x)^n / n! is the g.f. of A361551.
(5) a(n) = A361551(n-1)/(4*n-3) for n >= 1.
EXAMPLE
G.f.: A(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + 457227585*x^29 + ... + a(n)*x^(4*n-3) + ...
By definition, A(x - x^5*A'(x)) = x, where
A'(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + ... + A361551(n)*x^(4*n) + ...
Also,
A'(x) = 1 + (d/dx x^5*A'(x)) + (d^2/dx^2 x^10*A'(x)^2)/2! + (d^3/dx^3 x^15*A'(x)^3)/3! + (d^4/dx^4 x^20*A'(x)^4)/4! + (d^5/dx^5 x^25*A'(x)^5/5! + ... + (d^n/dx^n x^(5*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^4*A'(x) + (d/dx x^9*A'(x)^2)/2! + (d^2/dx^2 x^14*A'(x)^3)/3! + (d^3/dx^3 x^19*A'(x)^4)/4! + (d^4/dx^4 x^24*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A'(x)^n)/n! + ... ).
PROG
(PARI) {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^5*A' +x*O(x^(4*n)))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 17 2023
STATUS
approved