A307953 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(3*n) - A(x))^(n+1), where A(0) = 0.

1, 5, 16, 100, 745, 5981, 54668, 542147, 5770420, 65544681, 788168476, 9982471033, 132645367515, 1842830414090, 26692298441517, 402114353625235, 6287231891432992, 101837297768099079, 1705965231481768383, 29511920017674005949, 526496308467362015150, 9674316410154433376601, 182882315665489095973391, 3552928213442165146349142, 70865426835203730805138175

Offset

1,2

Links

Table of n, a(n) for n=1..25.

Formula

G.f. A(x) satisfies:

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

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

(3) 1 = Sum_{n>=0} x^n * (1-x)^(6*n+2) / ((1-x)^(3*n+1) - x*A(x/(1-x)))^(n+1).

(4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(3*n-3) * A(x/(1-x)))^n / (1-x)^(3*n^2-2*n-1)).

Example

G.f.: A(x) = x + 5*x^2 + 16*x^3 + 100*x^4 + 745*x^5 + 5981*x^6 + 54668*x^7 + 542147*x^8 + 5770420*x^9 + 65544681*x^10 + 788168476*x^11 + 9982471033*x^12 + ...
such that
1 = (1 - A(x)) + x*((1+x)^3 - A(x))^2 + x^2*((1+x)^6 - A(x))^3 + x^3*((1+x)^9 - A(x))^4 + x^4*((1+x)^12 - A(x))^5 + x^5*((1+x)^15 - A(x))^6 + x^6*((1+x)^18 - A(x))^7 + x^7*((1+x)^21 - A(x))^8 + ...
also
1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^3*A(x))^2 + x^2*(1+x)^6/(1 + x*(1+x)^6*A(x))^3 + x^3*(1+x)^18/(1 + x*(1+x)^9*A(x))^4 + x^4*(1+x)^36/(1 + x*(1+x)^12*A(x))^5 + x^5*(1+x)^60/(1 + x*(1+x)^15*A(x))^6 + ...

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+x +x*O(x^#A))^(3*m) - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A307940, A307952, A307954, A307955.

Sequence in context: A370096 A196015 A332625 * A304762 A317460 A341436

Adjacent sequences: A307950 A307951 A307952 * A307954 A307955 A307956

Keyword

nonn

Author

Paul D. Hanna, May 07 2019

Status

approved