A376528 G.f. A(x) satisfies (3*A(x^9)/A(x) - 1)^3 = 9*A(x^3)^3/A(x)^3 - 1 - 9*x.

1, 1, 4, 20, 141, 1045, 8172, 66681, 560173, 4815143, 42154010, 374555139, 3369272574, 30623456589, 280807674622, 2594626014807, 24133855340952, 225796163598383, 2123515433280699, 20063236095299562, 190347878004477651, 1812680623454591643, 17320874611556481567, 166020895158801131827

Offset

0,3

Comments

Compare to the following cubic modular identity of Jacobi theta functions,

(3*H(q^9)/H(q) - 1)^3 = 9*H(q^3)^4/H(q)^4 - 1,

which holds for H(q) = theta_2(q), H(q) = theta_3(q), and H(q) = theta_4(q) (see Mathworld link).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..730

Weisstein, Eric W., Modular Equation. From MathWorld -- A Wolfram Web Resource.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 141*x^4 + 1045*x^5 + 8172*x^6 + 66681*x^7 + 560173*x^8 + 4815143*x^9 + 42154010*x^10 + ...
RELATED SERIES.
(3*A(x^9)/A(x) - 1)^3 = 8 - 36*x - 54*x^2 - 171*x^3 - 1809*x^4 - 12258*x^5 - 92529*x^6 - 751626*x^7 + ...
9*A(x^3)^3 / A(x)^3   = 9 - 27*x - 54*x^2 - 171*x^3 - 1809*x^4 - 12258*x^5 - 92529*x^6 - 751626*x^7 + ...
so that 9*A(x^3)^3 / A(x)^3 - (3*A(x^9)/A(x) - 1)^3 = 1 + 9*x.

Prog

(PARI) {a(n) = my(V=[1], A); for(m=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( 1 + 9*x + (3*subst(A, x, x^9)/A - 1)^3 - 9*subst(A, x, x^3)^3/A^3, #V-1)/9 ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A376525, A376526.

Sequence in context: A187116 A340903 A098541 * A004204 A160567 A034216

Adjacent sequences: A376525 A376526 A376527 * A376529 A376530 A376531

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2024

Status

approved