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 A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function. 5
 1, 2, 0, 0, 2, 0, 0, 0, 0, 4, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 8, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel) Michael Somos, Introduction to Ramanujan theta functions, 2019. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions. FORMULA Expansion of eta(q^2)^5 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36))^2 in powers of q. Euler transform of period 36 sequence [2, -3, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t). a(n) = (-1)^n * A258322(n). a(4*n) = a(n). a(3*n + 2) = a(4*n + 3) = a(8*n + 6) = a(9*n + 3) = a(9*n + 6) = 0. a(3*n + 1) = 2 * A122865(n). a(6*n + 4) = 2 * A122856(n). a(9*n) = A004018(n). a(12*n + 1) = 2 * A002175(n). a(2*n) = A028601(n). - Michael Somos, Jul 04 2015 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Jan 29 2024 EXAMPLE G.f. = 1 + 2*q + 2*q^4 + 4*q^9 + 4*q^10 + 4*q^13 + 2*q^16 + 4*q^18 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9], {q, 0, n}]; a[ n_] := Which[ n < 1, Boole[n == 0], Mod[n, 3] == 2, 0, True, 2 DivisorSum[ n, If[ Mod[n/#, 9] > 0, 1, 2] KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jul 04 2015 *) PROG (PARI) {a(n) = if( n<1, n==0, (n+1)%3 * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))}; (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A))^2, n))}; (PARI) {a(n) = if( n<1, n==0, n%3==2, 0, 2 * sumdiv(n, d, if(n\d%9, 1, 2) * kronecker( -4, d)))}; /* Michael Somos, Jul 04 2015 */ (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); (n%3 < 2) * 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Jul 04 2015 */ (Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[19]; CROSSREFS Cf. A002175, A004018, A019670, A028601, A122856, A122865, A258322. Cf. A000122, A000700, A010054, A121373. Sequence in context: A107497 A000095 A258322 * A243828 A034949 A263767 Adjacent sequences: A258031 A258032 A258033 * A258035 A258036 A258037 KEYWORD nonn AUTHOR Michael Somos, Jun 03 2015 STATUS approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)