A363105 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).

1, 7, 59, 538, 5149, 51059, 520035, 5407889, 57181230, 612910369, 6644662132, 72731584789, 802696690614, 8922392225233, 99798739026795, 1122441028044882, 12686176392341722, 144013323190860339, 1641303449002365323, 18772674107796041770, 215413772477355781876

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

(1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^(n+1).

(2) 5 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (5*A(x) + x^n)^n.

(3) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n-1).

(4) 5*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n+1).

(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^n ].

(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (5*A(x) + x^n)^n ].

(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^(n+1))^n ].

(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (5*A(x) + x^n)^(n+1).

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^n)^n.

(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^n.

a(n) = Sum_{k=0..n} A359670(n,k) * 5^k for n >= 0.

Example

G.f.: A(x) = 1 + 7*x + 59*x^2 + 538*x^3 + 5149*x^4 + 51059*x^5 + 520035*x^6 + 5407889*x^7 + 57181230*x^8 + 612910369*x^9 + 6644662132*x^10 + ...

Prog

(PARI) {a(n) = my(A=1, y=5); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=[1], y=5); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363104.

Cf. A363185.

Sequence in context: A218201 A015568 A322667 * A101487 A210397 A099659

Adjacent sequences: A363102 A363103 A363104 * A363106 A363107 A363108

Keyword

nonn

Author

Paul D. Hanna, May 21 2023

Status

approved