A380679 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^n * (1+x)^(n*(3*n+1)/2) / A(x)^(n*(n+1)/2).

1, 1, 2, 3, 7, 15, 48, 177, 792, 3985, 21616, 125845, 773128, 4995356, 33768501, 237936780, 1742684309, 13233752561, 103985325332, 843956110146, 7064063586175, 60895536312911, 539984607186223, 4919957726789582, 46013558733949708, 441315845699885259, 4336873914330888090, 43633249440459934580

Offset

0,3

Comments

Compare to F(x) = Sum_{n>=0} x^n * (1+x)^(n*(3*n+1)/2) / F(x)^(3*n*(n+1)/2) holds when F(x) = (1+x).

Links

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 15*x^5 + 48*x^6 + 177*x^7 + 792*x^8 + 3985*x^9 + 21616*x^10 + 125845*x^11 + 773128*x^12 + ...
where
A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^7/A(x)^3 + x^3*(1+x)^15/A(x)^6 + x^4*(1+x)^26/A(x)^10 + x^5*(1+x)^40/A(x)^15 + x^6*(1+x)^57/A(x)^21 + x^7*(1+x)^77/A(x)^28 + x^8*(1+x)^100/A(x)^36 + ... + x^n * (1+x)^(n*(3*n+1)/2) / A(x)^(n*(n+1)/2) + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=0, #A, x^m*(1+x +x*O(x^#A))^(m*(3*m+1)/2) / A^(m*(m+1)/2)) - A, #V-1) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A321099.

Sequence in context: A045629 A034731 A216435 * A110888 A133736 A058698

Adjacent sequences: A380676 A380677 A380678 * A380680 A380681 A380682

Keyword

nonn

Author

Paul D. Hanna, Feb 22 2025

Status

approved