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Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.
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7
1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548
FORMULA
Expansion of eta(q^2) * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + u1 + u2) * (u3 + u6 + 3 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132976. G.f.: x * Product_{k>0} P(3,x^k) * P(9,x^k) * P(12,x^k) * P(36,x^k) where P(n,x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) * (1+x^(18*k)) / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)), n))};
Expansion of f(-x, -x^5) * f(-x)^2 / f(-x^6)^3 in powers of x where f(, ) and f() are Ramanujan theta functions.
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6
1, -3, 1, 3, -1, 0, 1, -6, 0, 6, -3, 3, 4, -12, 1, 12, -6, 3, 5, -24, 1, 24, -10, 6, 11, -42, 4, 42, -19, 12, 17, -72, 4, 69, -31, 18, 31, -120, 9, 114, -50, 30, 46, -189, 11, 180, -79, 48, 77, -294, 21, 276, -122, 72, 112, -450, 28, 420, -183, 108, 173, -672
FORMULA
Expansion of q^(1/3) * eta(q)^3 / (eta(q^2) * eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -2, -2, -2, -3, 0, ...].
Given g.f. A(x), then B(q) = A(q^3) / (3*q) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132302.
EXAMPLE
G.f. = 1 - 3*x + x^2 + 3*x^3 - x^4 + x^6 - 6*x^7 + 6*x^9 - 3*x^10 + 3*x^11 + ...
G.f. = 1/q - 3*q^2 + q^5 + 3*q^8 - q^11 + q^17 - 6*q^20 + 6*q^26 - 3*q^29 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x]^2 / QPochhammer[ x^6]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A)), n))};
Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM theta function.
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6
1, -3, 3, -3, 6, -9, 12, -15, 21, -30, 36, -45, 60, -78, 96, -117, 150, -189, 228, -276, 342, -420, 504, -603, 732, -885, 1050, -1245, 1488, -1773, 2088, -2454, 2901, -3420, 3996, -4662, 5460, -6378, 7404, -8583, 9972, -11565, 13344, -15378, 17748, -20448
COMMENTS
For n >= 1, a(n)/3 is a weighted count of overpartitions with restricted odd differences. Namely, the number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is odd and overlined. - Jeremy Lovejoy, Aug 07 2015
FORMULA
Expansion of chi(-q)^3 / chi(-q^3) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, 0, ...].
G.f.: Product_{k>0} (1 - x^(2*k-1))^3 / (1 - x^(6*k-3)).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - u * (2 - u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 2*u + 4) - v^3 * (u^2 + u + 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * (u6^2 - u2 * u3) - u6 * (u3^2 - u6*u2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A092848. Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = (-2+2*3^(1/2))^(1/3). - Simon Plouffe, Feb 20 2011 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
EXAMPLE
G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 6*q^4 - 9*q^5 + 12*q^6 - 15*q^7 + 21*q^8 + ...
MAPLE
with(numtheory):
a:= proc(n) option remember:
`if`(n=0, 1, add(add(d*[0, -3, 0, -2, 0, -3]
[irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
end:
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^3 QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *) a[n_] := a[n] = If[n==0, 1, Sum[Sum[d{0, -3, 0, -2, 0, -3}[[Mod[d, 6]+1]], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)), n))};
Expansion of f(q, q^2) * f(-q^3) / f(-q^2)^2 in powers of q where f(, ), f() are Ramanujan theta functions.
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4
1, 1, 3, 1, 6, 3, 12, 5, 21, 10, 36, 15, 60, 26, 96, 39, 150, 63, 228, 92, 342, 140, 504, 201, 732, 295, 1050, 415, 1488, 591, 2088, 818, 2901, 1140, 3996, 1554, 5460, 2126, 7404, 2861, 9972, 3855, 13344, 5126, 17748, 6816, 23472, 8970, 30876, 11793, 40413
FORMULA
Expansion of eta(q^3)^3 / (eta(q) * eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132179. G.f.: Product_{k>0} (1 + x^k + x^(2*k))^2 / ( (1 + x^k)^2 * (1 - x^k + x^(2*k))).
EXAMPLE
G.f. = 1 + q + 3*q^2 + q^3 + 6*q^4 + 3*q^5 + 12*q^6 + 5*q^7 + 21*q^8 + 10*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^3] QPochhammer[ -q^2, q^3] QPochhammer[ q^3]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A)), n))};
Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
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3
1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548
FORMULA
Expansion of eta(q^2) * eta(q^3) * eta(q^12) * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, -2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, ...].
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
1 + q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(6*k)) * (1+x^(9*k))^5 * (1-x^(9*k))^3 / ((1-x^(4*k)) * (1+x^(3*k)) * (1-x^(36*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))}
Expansion of f(x, x^5) * f(-x^6) / f(x)^2 in powers of x where f() is a Ramanujan theta function.
+10
3
1, -1, 3, -5, 10, -15, 26, -39, 63, -92, 140, -201, 295, -415, 591, -818, 1140, -1554, 2126, -2861, 3855, -5126, 6816, -8970, 11793, -15372, 20007, -25857, 33356, -42771, 54734, -69683, 88530, -111968, 141312, -177642, 222842, -278557, 347484, -432095, 536230
FORMULA
Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^2)^4 in powers of q.
Euler transform of period 12 sequence [ -1, 3, -2, 2, -1, 2, -1, 2, -2, 3, -1, 0, ...].
EXAMPLE
G.f. = 1 - x + 3*x^2 - 5*x^3 + 10*x^4 - 15*x^5 + 26*x^6 - 39*x^7 + ...
G.f. = q - q^3 + 3*q^5 - 5*q^7 + 10*q^9 - 15*q^11 + 26*q^13 - 39*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^12] / (QPochhammer[ x^2] QPochhammer[ -x]), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^2 + A)^4, n))};
Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function.
+10
2
1, -1, 1, -1, 2, -3, 4, -5, 7, -10, 12, -15, 20, -26, 32, -39, 50, -63, 76, -92, 114, -140, 168, -201, 244, -295, 350, -415, 496, -591, 696, -818, 967, -1140, 1332, -1554, 1820, -2126, 2468, -2861, 3324, -3855, 4448, -5126, 5916, -6816, 7824, -8970, 10292, -11793, 13471, -15372, 17548, -20007
FORMULA
Expansion of 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.
Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A139032. G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is the n-th cyclotomic polynomial.
a(n) = - A124243(n) unless n=0. a(2*n) = A128129(n) = a(2*n) unless n=0.
EXAMPLE
G.f. = 1 - q + q^2 - q^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))};
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