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Search: id:A029838
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| A029838 |
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Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8. |
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4
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| 1, 1, -1, 0, 1, 0, -1, -1, 2, 1, -2, -1, 2, 1, -3, -1, 4, 2, -5, -2, 5, 2, -6, -3, 8, 4, -9, -4, 10, 4, -12, -6, 15, 7, -17, -7, 19, 8, -22, -10, 26, 12, -30, -13, 33, 14, -38, -17, 45, 21, -51, -22, 56, 24, -64, -29, 74, 33, -83, -36, 92, 40, -104, -46, 119, 53, -133, -58, 147, 63, -165, -73, 187, 83, -208, -90, 229, 99, -256
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OFFSET
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0,9
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eqs. (9.1),(9.3).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
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FORMULA
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Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2)) where f(u, v)= 4+v^4-u^4*v^2 . - Michael Somos Mar 02 2006
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^3)) where f(u, v)= u^4 -v^4 -4*u*v +u^3*v^3 . - Michael Somos Mar 02 2006
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)= 2 +w^2 -u^2*v*w . - Michael Somos Mar 02 2006
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)= u2^2 +u6^2 -u1*u2*u3*u6 . - Michael Somos Ma4 02 2006
G.f.: Product_{k>0} (1+x^(2k-1))/(1+x^(2k)) = (Sum_{k>0} x^((k^2-k)/2)) / (Sum_{k>0} x^(k^2-k)).
Expansion of q^(1/8)eta(q^2)^3/(eta(q)eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ 1, -2, 1, 0, ...].
G.f. A(x) satisfies A(x)^2 = (A(x^4)+2x/A(x^4))/A(x^2). - Michael Somos Mar 08 2004
Expansion of f(q) / f(-q^4) = phi(q) / psi(q) = psi(q) / psi(q^2) = phi(-q^2) / psi(-q) = chi(q) * chi(-q^2) = chi^2(q) * chi(-q) = chi^2(-q^2) / chi(-q) = (phi(q) / psi(q^2))^(1/2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f.: 1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...))).
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EXAMPLE
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1/q + q^7 - q^15 + q^31 - q^47 - q^55 + 2*q^63 + q^71 - 2*q^79 - q^87 + ...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, (1+x^k)^(-(-1)^k), 1+x*O(x^n)), n))
(PARI) {a(n)=local(A); if(n<0, 0, A=contfracpnqn(matrix(2, (sqrtint(8*n+1)+1)\2, i, j, if(i==1, x^(j-1), 1+if(j>1, x^(j-1))))); polcoeff(A[1, 1]/A[2, 1]+x*O(x^n), n))} /* Michael Somos Mar 02 2006 */
(PARI) {a(n)=local(A, m); if(n<0, 0, A=1+O(x); m=1; while( m<=n, m*=2; A=subst(A, x, x^2); A2=subst(A, x, x^2); A=sqrt((A2+2*x/A2)/A)); polcoeff(A, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A) / eta(x^4 + A)^2, n))}
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CROSSREFS
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A082303(n)=(-1)^n a(n). A029839 is convolution square.
Adjacent sequences: A029835 A029836 A029837 this_sequence A029839 A029840 A029841
Sequence in context: A085342 A025825 A082303 this_sequence A023132 A023124 A023120
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KEYWORD
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sign
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AUTHOR
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njas
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