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McKay-Thompson series of class 10E for the Monster group with a(0) = 2.
+10
8
1, 2, 1, 2, 2, -2, -1, 0, -4, -2, 5, 2, 0, 8, 2, -8, -3, -2, -14, -6, 14, 6, 4, 24, 12, -24, -11, -4, -40, -16, 38, 16, 5, 62, 24, -60, -24, -10, -94, -40, 91, 38, 18, 144, 62, -136, -57, -24, -214, -88, 201, 82, 30, 308, 122, -288, -117, -48, -440, -180, 410, 168, 74, 624, 262, -578, -238, -96, -874, -356, 804
COMMENTS
Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) ( A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) ( A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) ( A000700).
FORMULA
Expansion of q^(-1) * (psi(q) / psi(q^5))^2 in powers of q where psi() is a Ramanujan theta function.
Expansion of ((eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q))^2 in powers of q.
Euler transform of period 10 sequence [ 2, -2, 2, -2, 0, -2, 2, -2, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v) * (v - 1) - 4 * v * (u - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u - 5) * v * (v - 1) * (v - 5).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138517. G.f.: (1/x) * (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-2) where P(n,x) is the n-th cyclotomic polynomial.
EXAMPLE
G.f. = 1/q + 2 + q + 2*q^2 + 2*q^3 - 2*q^4 - q^5 - 4*q^7 - 2*q^8 + 5*q^9 + ...
MATHEMATICA
QP = QPochhammer; s = ((QP[q^2]/QP[q^10])^2*(QP[q^5]/QP[q]))^2 + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 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) / eta(x^10 + A))^2 * eta(x^5 + A) / eta(x + A))^2, n))};
Expansion of q * chi(-q) / chi(-q^5)^5 in powers of q where chi() is a Ramanujan theta function.
+10
6
1, -1, 0, -1, 1, 4, -4, -1, -3, 3, 12, -12, -2, -8, 8, 31, -30, -5, -20, 19, 72, -68, -12, -44, 41, 154, -144, -24, -90, 84, 312, -289, -48, -178, 164, 603, -554, -92, -336, 307, 1122, -1024, -168, -612, 557, 2024, -1836, -300, -1087, 983, 3552, -3206, -522, -1880, 1692, 6088, -5472, -886, -3180
FORMULA
Expansion of (1 - (phi(-q) / phi(-q^5))^2) / 4 in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q) * eta(q^10)^5) / (eta(q^2) * eta(q^5)^5) in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 2*u*v + 4*u*v^2.
G.f. A(x) satisfies A(x^2) = -A(x) * A(-x).
Euler transform of period 10 sequence [ -1, 0, -1, 0, 4, 0, -1, 0, -1, 0, ...].
G.f.: x * (Product_{k>=1} ((1 - x^k) * (1-x^(10*k))^5) / ((1 - x^(2*k)) * (1 - x^(5*k))^5)).
Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -13/8 - (5/8)*sqrt(5) + (5/8)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 01 2021
EXAMPLE
q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^5 / (eta(x^2 + A) * eta(x^5 + A)^5), n))}
Expansion of chi(-q^5) / chi(-q)^5 in powers of q where chi() is a Ramanujan theta function.
+10
3
1, 5, 15, 40, 95, 205, 420, 820, 1535, 2785, 4915, 8460, 14260, 23590, 38360, 61440, 97055, 151370, 233355, 355900, 537395, 803960, 1192380, 1754140, 2560980, 3712205, 5344570, 7645600, 10871080, 15368350, 21607220, 30220360, 42056415, 58249680, 80310510
FORMULA
Expansion of (eta(q^5) / eta(q^10)) / (eta(q) / eta(q^2))^5 in powers of q.
Euler transform of period 10 sequence [ 5, 0, 5, 0, 4, 0, 5, 0, 5, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u * v * (2 - 4 * v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (1 + 3 * u - 4 * u^2) * (1 + 3 * v - 4 * v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132980. G.f.: Product_{k>0} (1 + x^k)^5 / (1 + x^(5*k)).
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = 1 + 5*q + 15*q^2 + 40*q^3 + 95*q^4 + 205*q^5 + 420*q^6 + 820*q^7 + ...
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[(1 + x^k)^5 / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2]^5, {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x^5 + A) / eta(x^10 + A) ) / ( eta(x + A) / eta(x^2 + A) )^5, n))};
McKay-Thompson series of class 20C for the Monster group with a(0) = -1.
+10
3
1, -1, 1, -2, 2, 2, -1, 0, -4, 2, 5, -2, 0, -8, 2, 8, -3, 2, -14, 6, 14, -6, 4, -24, 12, 24, -11, 4, -40, 16, 38, -16, 5, -62, 24, 60, -24, 10, -94, 40, 91, -38, 18, -144, 62, 136, -57, 24, -214, 88, 201, -82, 30, -308, 122, 288, -117, 48, -440, 180, 410
FORMULA
Expansion of q^(-1) * chi(q^5)^5 / chi(q) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^4) * eta(q^10)^10 / (eta(q^2)^2 * eta(q^5)^5 * eta(q^20)^5) in powers of q.
Euler transform of period 20 sequence [ -1, 1, -1, 0, 4, 1, -1, 0, -1, -4, -1, 0, -1, 1, 4, 0, -1, 1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u + 4) * v * (v - 1) * (v + 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225701. - Michael Somos, Sep 04 2013 G.f.: (1/x) * Product_{k>0} (1 + x^(10*k - 5))^5 / (1 + x^(2*k - 1)).
EXAMPLE
G.f. = 1/q - 1 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ -q^5, q^10]^5 / QPochhammer[ -q, q^2], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A)^10 / (eta(x^2 + A)^2 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};
McKay-Thompson series of class 20C for the Monster group with a(0) = 3.
+10
3
1, 3, 1, -2, 2, 2, -1, 0, -4, 2, 5, -2, 0, -8, 2, 8, -3, 2, -14, 6, 14, -6, 4, -24, 12, 24, -11, 4, -40, 16, 38, -16, 5, -62, 24, 60, -24, 10, -94, 40, 91, -38, 18, -144, 62, 136, -57, 24, -214, 88, 201, -82, 30, -308, 122, 288, -117, 48, -440, 180, 410
FORMULA
Expansion of q^(-1) * f(q)^3 / (f(-q^2) * f(-q^5)* f(-q^20)) in powers of q where f() is a Ramanujan theta function.
Expansion of q^(-1) * f(q, q)^2 / (f(q, q^9) * f(q^3, q^7)) in powers of q where f(, ) is Ramanujan's general theta functions.
Expansion of q^(-1) * (phi(q) / phi(q^5))^2 * chi(q^5)^5 / chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^8 / ( eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 3, -5, 3, -2, 4, -5, 3, -2, 3, -4, 3, -2, 3, -5, 4, -2, 3, -5, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 4) * (u - 5) * v * (v - 4) * (v - 5).
G.f.: (1/x) * Product_{k>0} (1 - (-x)^k)^3 / ((1 - x^(2*k)) * (1 - x^(5*k)) * (1 - x^(20*k))).
EXAMPLE
G.f. = 1/q + 3 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 4 + (1/q) QPochhammer[ q^5, q^10]^5 / QPochhammer[ q, q^2], {q, 0, n}];
a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ -q]^3 / (QPochhammer[ q^2] QPochhammer[ q^5] QPochhammer[ q^20]), {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / ( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};
McKay-Thompson series of class 10E for the Monster group with a(0) = -3.
+10
2
1, -3, 1, 2, 2, -2, -1, 0, -4, -2, 5, 2, 0, 8, 2, -8, -3, -2, -14, -6, 14, 6, 4, 24, 12, -24, -11, -4, -40, -16, 38, 16, 5, 62, 24, -60, -24, -10, -94, -40, 91, 38, 18, 144, 62, -136, -57, -24, -214, -88, 201, 82, 30, 308, 122, -288, -117, -48, -440, -180, 410
FORMULA
Expansion of eta(q)^3 * eta(q^5) / eta(q^2) / eta(q^10)^3 in powers of q.
Expansion of q^(-1) * phi(-q) * f(-q) / (psi(q^5) * f(-q^10)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 10 sequence [ -3, -2, -3, -2, -4, -2, -3, -2, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 20 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A095846. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + 4) * (20 + 6*v) - (v + 4) * (20 + v - u^2).
G.f.: (1 / x) * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) / ((1 - x^(2*k)) * (1 - x^(10*k))^3).
EXAMPLE
G.f. = 1/q - 3 + q + 2*q^2 + 2*q^3 - 2*q^4 - q^5 - 4*q^7 - 2*q^8 + 5*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ -4 + (1/q) QPochhammer[ q^5, q^10]^5 QPochhammer[ -q, q], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *) a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ q]^3 QPochhammer[ q^5] / (QPochhammer[ q^2] QPochhammer[ q^10]^3), {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) / eta(x^2 + A) / eta(x^10 + A)^3, n))};
Expansion of 1 / k(q) = 1 / (r(q) * r(q^2)^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.
+10
2
1, 1, 2, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 4, 1, -2, -6, -8, -7, -3, 4, 10, 14, 12, 6, -6, -16, -22, -20, -8, 8, 26, 34, 31, 12, -14, -41, -54, -47, -20, 23, 61, 84, 72, 31, -32, -90, -122, -107, -44, 45, 133, 174, 154, 61, -68, -192, -254, -220, -90, 100
COMMENTS
Number 12 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Aug 07 2014 A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(10). [Yang 2004] - Michael Somos, Aug 07 2014
FORMULA
Expansion of (1/x) * (f(-x^4, -x^6) * f(-x^3, -x^7)) / (f(-x^2, -x^8) * f(-x, -x^9)) in powers of x where f(,) is Ramanujan's two-variable theta function.
Euler transform of period 10 sequence [ 1, 1, -1, -1, 0, -1, -1, 1, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (u^2 - 1).
G.f.: (1/x) * Product_{k>0} (1 - x^(10*k - 3)) * (1 - x^(10*k - 4)) * (1 - x^(10*k - 6)) * (1 - x^(10*k - 7)) /((1 - x^(10*k - 1)) * (1 - x^(10*k - 2)) * (1 - x^(10*k - 8)) * (1 - x^(10*k - 9))).
EXAMPLE
G.f. = 1/q + 1 + 2*q + q^2 + q^3 - q^5 - 2*q^6 - 2*q^7 - q^8 + q^9 + 3*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q Product[(1 - q^k)^{-1, -1, 1, 1, 0, 1, 1, -1, -1, 0}[[Mod[k, 10, 1]]], {k, n + 1}], {q, 0, n}]; (* Michael Somos, Aug 07 2014 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, (1 - x^k + A)^[0, -1, -1, 1, 1, 0, 1, 1, -1, -1][k%10 + 1]), n))};
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