Search: a005798 -id:a005798
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A000521
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Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau).
(Formerly M5477 N2372)
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+10
334
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1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075
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OFFSET
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-1,2
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COMMENTS
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"The most natural normalization [of the j function] is to set the constant term equal to 24, the number given by Rademacher's infinite series for coefficients of the j function". [Borcherds]
Changing the term 744 to 24 gives A007240, the McKay-Thompson series of class 1A for Monster simple group.
sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
Klein's absolute invariant J=j/1728 is Gamma-modular.
The Mathematica implementation of KleinInvariantJ[] (versions 6 to 8) had bugs giving wrong value for a[7], a[9], a[11] and other values. - Michael Somos, Mar 07 2012
It is an open question if there are infinitely many k such that a(k) is prime. The known such indices are listed in A339429. See the paper by Fredrik Johansson. - Peter Luschny, May 05 2021
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REFERENCES
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J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 115.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, pp. 376ff.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 20.
Evans, David E., and Yasuyuki Kawahigashi. "Subfactors and mathematical physics." Bulletin of the American Mathematical Society, 60:4, (2023), 459-482 (see page 472).
M. Kaneko, Fourier coefficients of the elliptic modular function j(tau) (in Japanese), Rokko Lectures in Mathematics 10, Dept. Math., Faculty of Science, Kobe University, Rokko, Kobe, Japan, 2001.
M. J. Knopp, Rademacher on J(tau), Poincare series of nonpositive weights and Eichler cohomology, Notices Amer. Math. Soc., 37:4 (1990), 385-393.
S. Lang, Introduction to Modular Forms, Springer-Verlag, 1976, p. 12.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 56.
J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, see p. 482.
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).
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LINKS
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D. Alexander, C. Cummins, J. McKay, and C. Simons, Completely replicable functions, in Groups, Combinatorics & Geometry, (Durham, 1990), pp. 87--98, London Math. Soc. Monograph No. 165.
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
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FORMULA
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G.f.: A007245(q)^3/q; or (1 + 240 Sum_{k>0} sigma_3(k) q^k )^3 / (q Product_{k>0} (1-q^k)^24 ).
Expansion of 128 * (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) * (theta_2(q)^-8 + theta_3(q)^-8 + theta_4(q)^-8) in powers of q^2. - Michael Somos, Oct 02 2007
a(n) ~ exp(4*Pi*n^(1/2))/(2^(1/2)*n^(3/4)) [Petersson (1932), Rademacher (1938)]. - Gheorghe Coserea, Oct 09 2015
G.f.: 256*(1 - lambda + lambda^2)^3/(lambda^2 * (1 - lambda)^2) where lambda is the elliptic modular function (A115977). - Seiichi Manyama, Jul 30 2017
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EXAMPLE
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j = 1/q + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + ...
If J_n := j(sqrt(-n))^(1/3), then J_1 = 12, J_2 = 20, J_4 = 66, J_77 = 255. - Michael Somos, Oct 31 2019
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MAPLE
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with(numtheory): TOP := 31;
g2 := (4/3) * (1 + 240 * add(sigma[ 3 ](n)*q^n, n=1..TOP-1));
g3 := (8/27) * (1 - 504 * add(sigma[ 5 ](n)*q^n, n=1..TOP-1));
delta := series(g2^3 - 27*g3^2, q, TOP);
j := series(1728 * g2^3 / delta, q, TOP);
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MATHEMATICA
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CoefficientList[Normal[Series[1728*KleinInvariantJ[z], {z, 0, 30}]*Exp[ -2*I*Pi/z]] /. E^(Pi*Complex[0, n_]/z) -> t^(-n/2), t] (* Artur Jasinski, Dec 20 2008, after Daniel Lichtblau, corrected by Vaclav Kotesovec, Jul 07 2020 *)
a[ n_] := With[ {tau = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ 1728 KleinInvariantJ[ tau], {q, 0, n}], {q, 0, n}]]; (* Michael Somos, Nov 20 2011 *)(* Since V7 *)
a[ n_] := With[ {e1 = DedekindEta[ Log[q] / (2 Pi I)]^24, e2 = DedekindEta[ Log[q] / (Pi I)]^24}, SeriesCoefficient[ Series[ (e1 + 256 e2)^3 / (e1^2 e2), {q, 0, n + 1}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)
a[ n_] := With[ {L = ModularLambda[ Log[q] / (2 Pi I)]}, SeriesCoefficient[ Series[ 256 (L^2 - L + 1)^3 / (L (1 - L))^2, {q, 0, 2 n + 3}], {q, 0, n}]]; (* Michael Somos, Mar 09 2012 *)
a[ n_] := If[ n < -1, 0, With[ {E4 = 1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n + 2}], E6 = 1 - 504 Sum[ DivisorSigma[ 5, k] q^k, {k, n + 2}]}, SeriesCoefficient[ Series[ 1728 E4^3 / (E4^3 - E6^2), {q, 0, n}], {q, 0, n}]]]; (* Michael Somos, Mar 09 2012 *)
CoefficientList[Series[(65536 + x*QPochhammer[-1, x]^24)^3 / (16777216 * QPochhammer[-1, x]^24), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 23 2017 *)
a[n_] := SeriesCoefficient[With[{L = InverseEllipticNomeQ[rootQ]}, 256 (L^2 - L + 1)^3/(L (1 - L))^2], {rootQ, 0, 2n}]; (* Jan Mangaldan, Jul 07 2020, after Michael Somos; corrected by Leo C. Stein, Feb 25 2024 *)
a[n_] := SeriesCoefficient[ 12^3 KleinInvariantJ[Log[q]/(2 Pi I)], {q, 0, n}] (* Leo C. Stein, Feb 25 2024 *)
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PROG
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(PARI) {a(n) = my(A); if( n<-1, 0, A = x^(2*n + 2) * O(x); A = x * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8; polcoeff( subst( 256 * (1 - x + x^2)^3 / (x - x^2)^2, x, 16*A), 2*n))}; /* Michael Somos, Apr 30 2004 */
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^(5*n + 5) * O(x); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( subst( (x^2 + 10*x + 5)^3 / x, x, A), 5*n))}; /* Michael Somos, Apr 30 2004 */
(PARI) {a(n) = my(A); if( n<-1, 0, A = x^2 * O(x^n); A = x * (eta(x^2 + A) / eta(x + A))^24; polcoeff( (1 + 256*A)^3 / A, n))}; /* Michael Somos, Jul 13 2004 */
(PARI) q='q+O('q^66); Vec(ellj(q)) \\ Joerg Arndt, Apr 24 2016
(PARI) {a(n) = if( n<-1, 0, polcoeff( ellj(x + x^3 * O(x^n)), n))}; /* Michael Somos, Dec 25 2016 */
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CROSSREFS
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KEYWORD
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easy,nonn,nice,core
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AUTHOR
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EXTENSIONS
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Expanded the definition to include additional search terms. - N. J. A. Sloane, Nov 30 2019
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STATUS
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approved
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A115977
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Expansion of elliptic modular function lambda in powers of the nome q.
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+10
22
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16, -128, 704, -3072, 11488, -38400, 117632, -335872, 904784, -2320128, 5702208, -13504512, 30952544, -68901888, 149403264, -316342272, 655445792, -1331327616, 2655115712, -5206288384, 10049485312, -19115905536, 35867019904, -66437873664
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OFFSET
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1,1
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COMMENTS
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, eq. (37).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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Expansion of Jacobi elliptic parameter m = k^2 = (theta_2(q) / theta_3(q))^4 in powers of the nome q.
Expansion of 16 * q * (psi(q^2) / phi(q))^4 = 16 * q * (psi(q^2) / psi(q))^8 = 16 * q * (psi(q) / phi(q))^8 = 16 * q * (psi(-q) / phi(-q^2))^8 = 16 * q / (chi(q) * chi(-q^2))^8 = 16 * q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of 16 * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^8 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - v)^2 - 16 * v * (1 - u).
lambda( -1 / tau ) = 1 - lambda( tau ) (see A128692).
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692.
G.f.: 16 * q * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^8.
Empirical: Sum_{n>=1}(exp(-2*Pi)^n*a(n)) = 17 - 12*sqrt(2). - Simon Plouffe, Feb 20 2011
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)) / (32 * n^(3/4)). - Vaclav Kotesovec, Apr 06 2018
The g.f. A(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... satisfies A(q) + A(-q) = A(q)*A(-q). - Peter Bala, Sep 26 2023
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EXAMPLE
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G.f. = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + 11488*q^5 - 38400*q^6 + 117632*q^7 - ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ @ x, {x, 0, n}];
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ ModularLambda[ Log[q] / (Pi I)], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/16 (EllipticTheta[ 2, 0, q] / EllipticTheta[ 3, 0, q^2])^8, {q, 0, n}]; (* Michael Somos, May 26 2016 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); 16 * polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A005797
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Expansion of Jacobi nome q in terms of parameter m/16.
(Formerly M4561)
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+10
13
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0, 1, 8, 84, 992, 12514, 164688, 2232200, 30920128, 435506703, 6215660600, 89668182220, 1305109502496, 19138260194422, 282441672732656, 4191287776164504, 62496081197436736, 935823746406530603, 14065763582458332888, 212122153814497767004, 3208590886304243284640
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OFFSET
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0,3
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COMMENTS
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The Ansatz technique of A308835, A308836, and A308837 also works to produce the coefficients of this sequence from the ODE: T-d/dx(4*(1-x)*x*dT/dx)=0. - Bradley Klee, Jul 03 2019
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
C. L. Mallows, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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G.f.: q = q(m) = Sum_{n>=0} a(n) * (m/16)^n.
G.f.: exp( -Pi * agm(1, sqrt(1 - 16 * x) ) / agm(1, sqrt( 16*x ) ) ).
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EXAMPLE
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G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...
Given g.f. A(x), then q = exp(-Pi sqrt(6)) = A( m/16 ) where m = ((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2. - Michael Somos, Oct 30 2019
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MAPLE
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a:= n-> coeff(series(EllipticNome(4*sqrt(x)), x, n+1), x, n):
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 , n}] (* Michael Somos, Jul 11 2011 *)
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PROG
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(PARI) {a(n) = if( n < 1, 0, polcoeff( serreverse( x * prod(k=1, n-1, (1 + x^k)^(-1)^k, 1 + x * O(x^n))^8), n))} /* Michael Somos, Jul 19 2002 */
(PARI) {a(n) = my(A, m); if( n < 1, 0, m=1; A = x + O(x^2); while( m < n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff( serreverse(A), n))} /* Michael Somos, Mar 18 2003 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A014972
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Expansion of (theta_3(q) / theta_4(q) )^4 in powers of q; also of 1 / (1 - lambda(z)).
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+10
11
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1, 16, 128, 704, 3072, 11488, 38400, 117632, 335872, 904784, 2320128, 5702208, 13504512, 30952544, 68901888, 149403264, 316342272, 655445792, 1331327616, 2655115712, 5206288384, 10049485312, 19115905536, 35867019904, 66437873664
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OFFSET
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0,2
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COMMENTS
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The relation with A092877 is equivalent to eta(q^2)^24 = eta(q)^16 * eta(q^4)^8 + 16 * eta(q)^8 * eta(q^4)^16. - Michael Somos, Apr 11 2004
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
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LINKS
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FORMULA
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Expansion of 1 / (1 - lambda(t)) = 1 / lambda(-1 / t) in powers of q = exp(Pi i t).
Expansion of (phi(q) / phi(-q))^4 = (phi(-q^2) / phi(-q))^8 = (phi(q) / phi(-q^2))^8 = (f(q) / f(-q))^8 = (chi(q)/ chi(-q))^8 = (psi(q) / psi(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (eta(q^2)^3 / (eta(q^4) * eta(q)^2))^8 in powers of q. - Michael Somos, Apr 11 2004
Euler transform of period 4 sequence [ 16, -8, 16, 0, ...]. - Michael Somos, Apr 11 2004
G.f. A(x) satisfies A(-x) = 1 / A(x). Also 0 = f(A(x), A(x^2)) where f(u, v) = (u - 1)^2 + 16 * u*v * (1 - v). - Michael Somos, Apr 11 2004
G.f.: (Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(2*k - 1)))^8 = exp( 16 * Sum_{k>0} x^(2*k - 1) * sigma(2*k - 1) / (2*k - 1)). - Michael Somos, Apr 11 2004
Empirical : Sum_{n >=1} exp(-2*Pi)^(n-1)*(-1)^(n+1)*a(n) = -16+12*2^(1/2). - Simon Plouffe, Feb 20 2011
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EXAMPLE
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G.f. = 1 + 16*q + 128*q^2 + 704*q^3 + 3072*q^4 + 11488*q^5 + 38400*q^6 + 117632*q^7 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^4, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / (QPochhammer[ q^4] QPochhammer[ q]^2))^8, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( exp( 16 * sum( k=1, (n+1)\2, sigma(2*k - 1) / (2*k - 1) * x^(2*k - 1), x * O(x^n))), n))}; /* Michael Somos, Apr 11 2004 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^8, n))}; /* Michael Somos, Apr 11 2004 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A092877
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Expansion of (eta(q^4) / eta(q))^8 in powers of q.
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+10
10
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1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
G.f.: theta_2^4 / (16*theta_4^4) = lambda / (16 * (1 - lambda)).
G.f.: exp( Integral theta_3(x)^4/x dx ). - Paul D. Hanna, May 03 2010
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EXAMPLE
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G.f. = q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
eta[q_]:= q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^4] / eta[q])^8, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* Vincenzo Librandi, Oct 18 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( x * prod(k=1, (n+1)\2, (1 + x^(2*k)) / (1 - x^(2*k-1)), 1 + x * O(x^n))^8, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};
(PARI) a(n)= { local(A); n--; A=x*O(x^n); polcoeff((eta(x^4 + A)/eta(x + A))^8, n); } { for(n=1, 1000, write("b092877.txt", n, " ", a(n)); ); } \\ Harry J. Smith, Jun 21 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A225915
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Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.
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+10
1
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1, -16, 152, -1088, 6444, -33184, 153152, -646528, 2533070, -9311664, 32387616, -107299904, 340436664, -1039026144, 3061896704, -8739810688, 24229115109, -65390485328, 172155210320, -442928464640, 1115433685796, -2753362613984, 6670224790272, -15876957230848
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OFFSET
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2,2
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LINKS
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FORMULA
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Expansion of (eta(q) * eta(q^4)^2 / eta(q^2)^3)^16 in powers of q.
Euler transform of period 4 sequence [-16, 32, -16, 0, ...].
G.f.: q^2 * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^16.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (65536 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017
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EXAMPLE
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G.f. = q^2 - 16*q^3 + 152*q^4 - 1088*q^5 + 6444*q^6 - 33184*q^7 + 153152*q^8 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ (InverseEllipticNomeQ[ q] / 16)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^16, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q ( Product[ 1 - q^k, {k, 4, n - 1, 4}]/
Product[ 1 - (-q)^k, {k, n - 1}])^16, {q, 0, n}];
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PROG
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(PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^16, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A275790
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Triangle T(n, m) appearing in the expansion of the scaled phase space coordinate qhat of the plane pendulum in terms of the Jacobi nome q and sin(v) multiplying even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).
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+10
1
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1, 8, 1, -32, 11, 3, -736, -92, 9, 15, 2816, -593, -249, -65, 35, 48976, 6122, 1581, -970, -1295, 315, -951424, 61252, 67791, 46030, 18515, -21735, 3465, -1045952, -130744, -92082, -30445, 14455, 53928, -25179, 3003, 26933248, 1069361, -1666047, -634255, -1167740, -1258236, 1562253, -471471, 45045, 634836808, 79354601, 24881793, 17914550, 30289840, 12635028, -71064609, 42480438, -9594585, 765765
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OFFSET
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0,2
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COMMENTS
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The dimensionless scaled phase space coordinates of the plane pendulum are (qtilde(tau, k), ptilde(tau, k)) with tau = omega_0*t, omega^2 = g/L (L is the length of the pendulum, g the acceleration), and the energy variable E = 2*k^2 = 2*sin^2(Theta_0/2), with the maximal deflection angle Theta_0 (from [0, Pi/2]). qtilde = Theta/(2*k)) with the deflection angle Theta. Similarly ptilde = (d(Theta)/d(tau))/(2*k).
The exact solution is qtilde(tau, k) = (1/k)*arcsin(k*sn(tau, k)) with Jacobi's elliptic sn function, and ptilde(tau,k) = cn(tau, k) with the elliptic cn function.
Here the expansion in new variables v and q is used where v = tau/((2/Pi)*K(k)) and q = exp(-Pi*K'(k)/K(k)) with the real and imaginary quarter periods K and K'. This leads to qhat(v, q) = qtilde(tau(v, q), k(q)) with tau(v, q) = theta_3^2(0, q)*v. (For theta_3^2(0, q) see A004018.) Because k is actually a function of k^2 one uses the q expansion of (k/4)^2 given in A005798.
Using the result for the sn expansion in q and v from A274662 one obtains qhat(v, q) = sin(v)*Sum_{n >= 0} q^n/L(n)*Sum_{m=0..n} T(n, m)*(2*cos(v))^(2*m) with L(n) = A025547(n+1) = lcm{1, 3, ..., (2*n+1)}.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506). Thanks for discussions via e-mail go to him.
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LINKS
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FORMULA
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T(n, m)*(2*cos(v))^(2*m)), n >= 0, m = 0, 1, ..., n, gives the contribution to q^n/L(n) (L(n) = A025547(n+1)) in the rescaled phase space coordinate qhat(v, q) expansion of the plane pendulum. See a comment above for details.
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EXAMPLE
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The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 ...
0: 1
1: 8 1
2: -32 11 3
3: -736 -92 9 15
4: 2816 -593 -249 -65 35
5: 48976 6122 1581 -970 -1295 315
...
row n=6: -951424 61252 67791 46030 18515 -21735 3465,
row n=7: -1045952 -130744 -92082 -30445 14455 53928 -25179 3003,
row n=8: 26933248 1069361 -1666047 -634255 -1167740 -1258236 1562253 -471471 45045,
row n=9: 634836808 79354601 24881793 17914550 30289840 12635028 -71064609 42480438 -9594585 765765.
...
The corresponding L(n) = A025547(n+1) numbers are 1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535,...
n=4: the contribution to qhat(v, q) of order q^4 is (q^4/315)*(2816 - 593*(2*cos(v))^2 - 249*(2*cos(v))^4 - 65*(2*cos(v))^6 + 35*(2*cos(v))^8).
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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