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 A143752 Expansion of eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q. 3
 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 14, 17, 20, 23, 27, 31, 36, 41, 48, 55, 63, 72, 82, 94, 106, 122, 137, 156, 175, 197, 222, 249, 280, 314, 352, 393, 439, 490, 546, 608, 676, 751, 834, 923, 1024, 1133, 1253, 1384, 1528, 1686, 1857, 2045, 2250, 2474, 2718 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 Michael Somos, A Remarkable eta-product Identity FORMULA Expansion of F(q) * F(q^2) in powers of q^3 where F(q) is the g.f. for A103263. Euler transform of a period 60 sequence. G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143751. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 + v^2) * (1 + u + v) * (u + v + u*v) - u*v * (1+ 2*u + 2*v + u*v)^2. G.f.: x * Product_{k>0} P(30, x^k) * P(60, x^k) where P(n, x) is the n-th cyclotomic polynomial. a(2*n) = A123630(n). Convolution inverse of A143751. G.f.: -1 + Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))). - Seiichi Manyama, May 04 2017 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018 EXAMPLE G.f. = q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + 7*q^10 + ... PROG (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))}; CROSSREFS Cf. A123630, A143751, A145933. Sequence in context: A029023 A140952 A096911 * A145933 A300788 A120171 Adjacent sequences: A143749 A143750 A143751 * A143753 A143754 A143755 KEYWORD nonn AUTHOR Michael Somos, Aug 31 2008 STATUS approved

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Last modified March 1 20:05 EST 2024. Contains 370443 sequences. (Running on oeis4.)