A325584 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 4*x)^n - A(x))^(n+1), where A(0) = 0.

1, 7, 17, 143, 1297, 10943, 119041, 1352319, 16521601, 217712895, 3035672577, 44699885311, 692651630593, 11245459802111, 190749994213377, 3372199652642815, 61989222776496129, 1182514506870886399, 23367685697859391489, 477573301465741901823, 10079510247865215746049, 219396394337370417070079, 4918940850123829203173377, 113466581251217062104399871, 2690031991636202195545948161

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} x^n * ((1 + 4*x)^n - A(x))^(n+1).

(2) 1 + x = Sum_{n>=0} x^n * (1 + 4*x)^(n*(n-1)) / (1 + x*(1 + 4*x)^n*A(x))^(n+1).

FORMULA FOR TERMS.

a(n) = (-1)^n (mod 4) for n >= 0.

Example

G.f.: A(x) = x + 7*x^2 + 17*x^3 + 143*x^4 + 1297*x^5 + 10943*x^6 + 119041*x^7 + 1352319*x^8 + 16521601*x^9 + 217712895*x^10 + 3035672577*x^11 + ...
such that
1 = (1 - A(x)) + x*((1+4*x) - A(x))^2 + x^2*((1+4*x)^2 - A(x))^3 + x^3*((1+4*x)^3 - A(x))^4 + x^4*((1+4*x)^4 - A(x))^5 + x^5*((1+4*x)^5 - A(x))^6 + x^6*((1+4*x)^6 - A(x))^7 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1 + 4*x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A307940, A325582, A325583, A325585.

Sequence in context: A227506 A244279 A364703 * A214149 A147643 A367809

Adjacent sequences: A325581 A325582 A325583 * A325585 A325586 A325587

Keyword

nonn

Author

Paul D. Hanna, May 11 2019

Status

approved