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Search: a102314 -id:a102314
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Expansion of 1 / (chi(-x) * chi(-x^7)) in powers of x where chi() is a Ramanujan theta function.
+10
5
1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5962, 6736, 7606, 8582, 9666, 10884
OFFSET
0,4
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the right side of Cayley's identity is 2 * q * A(q^2). - Michael Somos, Dec 03 2013
Proof of Cayley's identity, from Silviu Radu, Mar 13 2015: (Start)
Up to issues of convergence I observe that the identity may be rewritten after substituting q=e^{2 Pi Iz} as:
E(28z)^(-1) x E(14z)^2 x E(7z)^(-1) x E(4z)^(-1) x E(2z)^2 x E(z)^(-1) -E(14z)^(-1) x E(7z) x E(2z)^(-1) x E(z) = 2 E(28z) x E(14z)^(-1) x E(4z) x E(2z)^(-1)
where E(z)= exp( Pi I z/12) Product_{n>=1} (1-e^{2 Pi I z n}) is the Dedekind eta function.
One can further rewrite the above identity by dividing the whole identity by the first term. We obtain:
1-E(28z) x E(14z)^(-3) x E(7z)^2 x E(4z) x E(2z)^(-3) x E(z)^2
-2 E(28z)^2 x E(14z)^(-3) x E(7z) x E(4z)^2 x E(2z)^(-3) x E(z)=0
What is interesting now about this expression is that each term is a modular function for the group Gamma_0(28).
Furthermore, all the terms except the constant term have two poles, therefore the whole left hand side has at most two poles (at the points z=1/14 and z=1/2).
However we check that in the q-expansion the first three coefficients are zero, which implies that the left hand side also has a zero of order at least three at the point infinity (note that z=I x infty transforms into q=0, q=e^(2 Pi iz} ).
It is impossible that a nonzero modular function has more zeros than poles, therefore it is the zero function. This finishes the proof. (End)
REFERENCES
A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
LINKS
FORMULA
Expansion of q^(-1/3) * (eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)) in powers of q.
Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 2*u*v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. of A102314. - Michael Somos, Dec 03 2013
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(7*k)).
a(n) = A112212(2*n + 1) = - A102314(2*n + 1). - Michael Somos, Dec 03 2013
Convolution inverse of A102314.
a(n) = (-1)^n * A246762(n). - Michael Somos, Sep 02 2014
a(n) ~ exp(2*Pi*sqrt(2*n/21)) / (2^(7/4) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 7*x^8 + ...
G.f. = q + q^4 + q^7 + 2*q^10 + 2*q^13 + 3*q^16 + 4*q^19 + 6*q^22 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^7, x^7], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + x^(7*k), 1 + x * O(x^n)), n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^14 + A) / (eta(x + A) * eta(x^7 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Apr 19 2004
EXTENSIONS
Entry revised by N. J. A. Sloane, Mar 15 2015 (with thanks to Doron Zeilberger)
STATUS
approved
McKay-Thompson series of class 84C for the Monster group.
+10
4
1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 3, 3, 4, 4, 4, 6, 7, 7, 7, 9, 10, 12, 13, 14, 17, 18, 19, 22, 26, 28, 29, 34, 38, 41, 44, 50, 57, 60, 65, 72, 81, 86, 94, 105, 114, 124, 133, 146, 161, 174, 187, 204, 224, 240, 258, 282, 309, 332, 354, 386, 419, 450, 481, 524, 569, 606
OFFSET
0,8
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the first term of the left side of Cayley's identity is A(q). - Michael Somos, Dec 03 2013
REFERENCES
A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
LINKS
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - Michael Somos, Dec 03 2013
Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - Michael Somos, Dec 03 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 03 2013
G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - Michael Somos, Dec 03 2013
a(n) = (-1)^n * A102314(n). a(2*n + 1) = A093950(n). - Michael Somos, Dec 03 2013
a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015
EXAMPLE
G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 3*x^10 + ...
T84C = 1/q + q^2 + q^8 + q^11 + q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^7, x^14], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Dec 03 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 28 2005
STATUS
approved
Expansion of psi(x^2) * phi(x^7) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
+10
2
1, 1, 3, 4, 7, 10, 17, 26, 38, 57, 81, 114, 161, 224, 309, 419, 569, 759, 1011, 1336, 1757, 2296, 2981, 3855, 4956, 6344, 8087, 10272, 12994, 16367, 20561, 25723, 32086, 39902, 49484, 61182, 75439, 92791, 113821, 139294, 170073, 207187, 251853
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(-x^14)^2 / (chi(-x) * chi(-x^2)^4 * chi(-x^7)^3 ) in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/12) * eta(q^4)^2 * eta(q^14)^5 / (eta(q) * eta(q^2) * eta(q^7)^3* eta(q^28)^2) in powers of q.
Euler transform of period 28 sequence [ 1, 2, 1, 0, 1, 2, 4, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 4, 2, 1, 0, 1, 2, 1, 0, ...].
a(n) = A102314(4*n).
a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
EXAMPLE
G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 26*x^7 + 38*x^8 + ...
G.f. = 1/q + q^11 + 3*q^23 + 4*q^35 + 7*q^47 + 10*q^59 + 17*q^71 + 26*q^83 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^7] EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^14, x^28]^2 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^7, x^14]^3), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^14 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^7 + A)^3* eta(x^28 + A)^2), n))};
CROSSREFS
Cf. A102314.
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 06 2011
STATUS
approved
Expansion of x * phi(x) * psi(x^14) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
+10
2
0, 1, 3, 4, 7, 13, 19, 29, 44, 65, 94, 133, 187, 258, 354, 481, 651, 871, 1154, 1526, 1998, 2603, 3376, 4358, 5594, 7148, 9103, 11531, 14560, 18320, 22972, 28708, 35757, 44413, 54990, 67904, 83626, 102736, 125890, 153882, 187694, 228396, 277336
OFFSET
0,3
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(-x^2)^2 / (chi(-x)^3 * chi(-x^7) * chi(-x^14)^4 ) in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(-5/12) * eta(q^2)^5 * eta(q^28)^2 / (eta(q)^3 * eta(q^4)^2 * eta(q^7) * eta(q^14)) in powers of q.
Euler transform of period 28 sequence [ 3, -2, 3, 0, 3, -2, 4, 0, 3, -2, 3, 0, 3, 0, 3, 0, 3, -2, 3, 0, 4, -2, 3, 0, 3, -2, 3, 0, ...].
a(n) = A102314(4*n + 2).
a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
EXAMPLE
G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 19*x^6 + 29*x^7 + 44*x^8 + 65*x^9 + ...
G.f. = q^17 + 3*q^29 + 4*q^41 + 7*q^53 + 13*q^65 + 19*q^77 + 29*q^89 + 44*q^101 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n + 2}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4]^2 / (QPochhammer[ x^7, x^14] QPochhammer[ x^14, x^28]^2 QPochhammer[ x, x^2]^3), {x, 0, n - 1}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^7] / (2 QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n + 3/4}];
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^28 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^7 + A) * eta(x^14 + A)), n))};
CROSSREFS
Cf. A102314.
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 07 2011
STATUS
approved

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