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Expansion of a modular function for Gamma_0(6).
(Formerly M1542 N0602)
+10
12
1, 2, -5, -24, -23, 76, 249, 168, -599, -1670, -1026, 3272, 8529, 5232, -14062, -35976, -22337, 51516, 131617, 82568, -169376, -432636, -273332, 513584, 1309800, 830372, -1456569, -3709672, -2354215, 3904696, 9931407, 6301120, -9983208, -25339626, -16057040, 24504584, 62033318
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
Expansion of q^-3 * psi(q)^6 * phi(-q)^2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 24 2014 Expansion of eta(q^2)^10 * eta(q^3)^14 / (eta(q)^2 * eta(q^6)^22) in powers of q.
Euler transform of period 6 sequence [2, -8, -12, -8, 2, 0, ...]. - Michael Somos, Nov 10 2005 Convolution product of A128632, A128633, and A105559 (all three of them are modular functions and McKay-Thompson series of class 6E for the monster group). - Michael Somos, May 23 2014
EXAMPLE
G.f. = q^-3 + 2*q^-2 - 5*q^-1 - 24 - 23*q + 76*q^2 + 249*q^3 + 168*q^4 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q^2]^10*(QP[q^3]^14/(QP[q]^2*QP[q^6]^22)) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
PROG
(PARI) {a(n) = local(A); if( n<-3, 0, n+=3; A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^3 + A)^14 / (eta(x + A)^2 * eta(x^6 + A)^22), n))}; /* Michael Somos, Nov 10 2005 */
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 14 2001
McKay-Thompson series of class 6E for the Monster group with a(0) = 4.
+10
8
1, 4, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744
FORMULA
Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^-1 * (psi(q) / psi(q^3))^4 in powers of q where psi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)^2))^4 in powers of q.
Euler transform of period 6 sequence [ 4, -4, 0, -4, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (u - 9) * (u - 1) - (u - v)^2.
G.f.: (1/x) * (Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^k + x^(2*k))^2)^-4.
EXAMPLE
G.f. = 1/q + 4 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)])^4, {q, 0, n}]; (* Michael Somos, Nov 12 2015 *) QP = QPochhammer; s = (QP[q^2]^2*(QP[q^3]/(QP[q]*QP[q^6]^2)))^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2))^4, n))};
Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(-q)^3 / phi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
+10
8
1, 5, 19, 61, 174, 455, 1112, 2573, 5689, 12102, 24900, 49759, 96902, 184408, 343722, 628717, 1130418, 2000669, 3489788, 6005910, 10207688, 17147892, 28494120, 46865519, 76342903, 123236446, 197233723, 313106264, 493231830, 771301986, 1197743552, 1847606573
FORMULA
Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q)^2 / b(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^6) / eta(q))^5 * eta(q^2) / eta(q^3) in powers of q.
Euler transform of period 6 sequence [ 5, 4, 6, 4, 5, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 8*u) * (1 + 9*v) - (u-v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/72) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128632. G.f.: x * Product_{k>0} ((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (72 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
EXAMPLE
G.f. = q + 5*q^2 + 19*q^3 + 61*q^4 + 174*q^5 + 455*q^6 + 1112*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^6] / QPochhammer[ q])^5 (QPochhammer[ q^2] / QPochhammer[ q^3]), {q, 0, n}]; (* Michael Somos, Jun 08 2015 *) nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(6*k)) / (1-x^k))^5 * ((1 - x^(2*k)) / (1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) / eta(x + A))^5 * eta(x^2 + A) / eta(x^3 + A), n))};
McKay-Thompson series of class 6E for the Monster group with a(0) = 1.
+10
4
1, 1, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744, -20448, -46944, -26916
FORMULA
Expansion of (1/q) * a(q^2) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function and a() is a cubic AGM theta function. - Michael Somos, May 22 2015 Expansion of 6 + eta(q)^5 * eta(q^3) / (eta(q^2) * eta(q^6)^5) in powers of q. - Michael Somos, May 22 2015
EXAMPLE
G.f. = 1/q + 1 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 6 + QPochhammer[ q]^5 QPochhammer[ q^3] / (q QPochhammer[ q^2] QPochhammer[ q^6]^5), {q, 0, n}]; (* Michael Somos, May 22 2015 *) a[ n_] := SeriesCoefficient[ -2 + (1/q) (QPochhammer[ q^2] QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^6]^3))^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3] + 3 EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q]) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)]^3, {q, 0, n}]; (* Michael Somos, May 22 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 6*x + eta(x + A)^5 * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)^5), n))}; /* Michael Somos, May 22 2015 */ (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -2*x + (eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3))^3, n))}; /* Michael Somos, May 22 2015 */
McKay-Thompson series of class 12B for the Monster group with a(0) = 5.
+10
4
1, 5, 6, -4, -3, 12, -8, -12, 30, -20, -30, 72, -46, -60, 156, -96, -117, 300, -188, -228, 552, -344, -420, 1008, -603, -732, 1770, -1048, -1245, 2976, -1776, -2088, 4908, -2900, -3420, 7992, -4658, -5460, 12756, -7408, -8583, 19944, -11564, -13344, 30756
FORMULA
Expansion of (1/q) * (phi(q)^3 * psi(-q)) / (phi(q^3) * psi(-q^3)^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^14 / (eta(q)^5 * eta(q^3) * eta(q^4)^5 * eta(q^6)^2 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 5, -9, 6, -4, 5, -6, 5, -4, 6, -9, 5, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A164617.
EXAMPLE
G.f. = 1/q + 5 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 - 20*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 2, Pi/4, q^(1/2)] / (EllipticTheta[ 3, 0, q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3), {q, 0, n}]; (* Michael Somos, May 20 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^14 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 * eta(x^12 + A)), n))};
McKay-Thompson series of class 6E for the Monster group with a(0) = 7.
+10
4
1, 7, 6, 4, -3, -12, -8, 12, 30, 20, -30, -72, -46, 60, 156, 96, -117, -300, -188, 228, 552, 344, -420, -1008, -603, 732, 1770, 1048, -1245, -2976, -1776, 2088, 4908, 2900, -3420, -7992, -4658, 5460, 12756, 7408, -8583, -19944, -11564, 13344, 30756, 17744
FORMULA
Expansion of q^(-1) * a(q) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function and a() is a cubic AGM theta function.
Expansion of 12 + eta(q)^5 * eta(q^3) / (eta(q^2) * eta(q^6)^5) in powers of q.
EXAMPLE
G.f. = 1/q + 7 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...
MATHEMATICA
QP = QPochhammer; s = 4*q + (QP[q^2]*(QP[q^3]^3/(QP[q]*QP[q^6]^3)))^3 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 4*x + (eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3))^3, n))};
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