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A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions. 1
1, 15, 81, 241, 624, 1215, 2402, 3855, 6561, 9360, 14640, 19521, 28562, 36030, 50544, 61681, 83520, 98415, 130322, 150384, 194562, 219600, 279840, 312255, 390001, 428430, 531441, 578882, 707280, 758160, 923522, 986895, 1185840, 1252800, 1498848, 1581201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

FORMULA

a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).

Convolution of A008653 and A106402.

EXAMPLE

G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...

MATHEMATICA

a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];

PROG

(PARI) {a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};

(PARI) {a(n) = my(A, p, e, s); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, p^(4*e), s=-(-1)^(p%3);  ((p^4)^(e+1) - s^(e+1)) / (p^4 - s))))};

(MAGMA) A := Basis( ModularForms( Gamma1(3), 5), 37);  A[2];

CROSSREFS

C.f. A008653, A106402.

Sequence in context: A244855 A102360 A309336 * A213552 A060581 A253222

Adjacent sequences:  A266285 A266286 A266287 * A266289 A266290 A266291

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Dec 26 2015

STATUS

approved

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Last modified January 27 21:53 EST 2020. Contains 331300 sequences. (Running on oeis4.)