A352815 G.f. A(x) satisfies: 1 + x = Sum_{n>=0} (-1)^n * (x^n + A(x))^(n+1).

1, 4, 15, 62, 263, 1153, 5187, 23792, 110898, 523773, 2501268, 12057407, 58593831, 286743949, 1411905287, 6989973590, 34773216944, 173737947911, 871442154413, 4386482848975, 22150822685669, 112185906664804, 569713055956736, 2900350345874632, 14799219791196091

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = x + 4*x^2 + 15*x^3 + 62*x^4 + 263*x^5 + 1153*x^6 + 5187*x^7 + 23792*x^8 + 110898*x^9 + 523773*x^10 + ...
where
1 + x = (1 + A(x)) - (x + A(x))^2 + (x^2 + A(x))^3 - (x^3 + A(x))^4 + (x^4 + A(x))^5 - (x^5 + A(x))^6 + (x^6 + A(x))^7 + ...
also
-x = 1/(1 + A(x)) - 1/(1 + x*A(x))^2 + x^2/(1 + x^2*A(x))^3 - x^6/(1 + x^3*A(x))^4 + x^12/(1 + x^4*A(x))^5 - x^20/(1 + x^5*A(x))^6 + ...
Specific values.
A(x) = 1 at x = 0.1834970136530040531685106821803389905413247357336272...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, (-1)^m*(x^m + x*Ser(A))^(m+1) ), #A)); A[n]}
for(n=1, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A)+1, (-1)^m*x^(m*(m-1))/(1 + x^m*x*Ser(A))^(m+1) ), #A)); A[n]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A317997.

Sequence in context: A151484 A275871 A007161 * A007167 A036728 A027216

Adjacent sequences: A352812 A352813 A352814 * A352816 A352817 A352818

Keyword

nonn

Author

Paul D. Hanna, Apr 04 2022

Status

approved