A352855 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n).

1, 1, 3, 13, 69, 418, 2785, 19965, 151839, 1213329, 10117007, 87572771, 783819258, 7231777871, 68607958358, 667912243459, 6661081903672, 67956552527140, 708354657022645, 7536107197624606, 81756574537050031, 903706858640897865, 10170700352770130132

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n),

(2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+3)).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 418*x^5 + 2785*x^6 + 19965*x^7 + 151839*x^8 + 1213329*x^9 + 10117007*x^10 + ...
where
(1) 1 = A(x) - x*A(x)^3*A(x*A(x)) + x^2*A(x)^6*A(x*A(x)^2) - x^3*A(x)^9*A(x*A(x)^3) + x^4*A(x)^12*A(x*A(x)^4) - x^5*A(x)^15*A(x*A(x)^5) + x^6*A(x)^18*A(x*A(x)^6) + ...
(2) 1 = 1/(1 + x*A(x)^3) + 1*x/(1 + x*A(x)^4) + 3*x^2/(1 + x*A(x)^5) + 13*x^3/(1 + x*A(x)^6) + 69*x^4/(1 + x*A(x)^7) + 418*x^5/(1 + x*A(x)^8) + 2785*x^6/(1 + x*A(x)^9) + ... + a(n)*x^n/(1 + x*A(x)^(n+3)) + ...

Prog

(PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(3*n) * A(x*A(x)^n) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, (-x)^n*Ser(A)^(3*n)*subst(Ser(A), x, x*Ser(A)^n) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+3)) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(n=0, #A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+3)) ), #A-1)); A[n+1]}
for(n=0, 31, print1(a(n), ", "))

Crossrefs

Cf. A352853, A352854, A352856.

Sequence in context: A153395 A243688 A368708 * A088714 A067145 A192739

Adjacent sequences: A352852 A352853 A352854 * A352856 A352857 A352858

Keyword

nonn

Author

Paul D. Hanna, Apr 05 2022

Status

approved