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A092876
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Expansion of q * (f(-q, -q^12) * f(-q^3, -q^10) * f(-q^4, -q^9)) / (f(-q^2, -q^11) * f(-q^5, -q^8) * f(-q^6, -q^7)) in powers of q where f(, ) is Ramanujan's general theta function.
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4
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1, -1, 1, -2, 1, 0, 1, 1, -1, -2, 0, 0, 2, 2, -3, 2, -6, 3, 1, 2, 2, -2, -4, 0, -2, 5, 7, -8, 6, -16, 7, 1, 6, 6, -7, -10, 1, -2, 11, 14, -17, 12, -34, 16, 3, 12, 11, -12, -22, 1, -6, 24, 30, -36, 25, -70, 32, 6, 25, 24, -26, -42, 2, -10, 45, 56, -68, 48, -132, 60, 12, 45, 43, -46, -78, 4, -22, 84, 106, -126, 89, -242, 110, 20, 84, 80
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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Euler transform of period 13 sequence [-1, 1, -1 ,-1, 1, 1 ,1, 1, -1, -1, 1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + u*v^3 + u^3*v^2 + 2*u*v * (1 - u + v + u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3*v * (3 + 3*v + v^2) + 3*u^2*v * (v^2 + v - 1) + u*v * (1 - 3*v + 3*v^2) - (u^4 + v^4)
G.f.: x * Product_{k>0} (1 - x^k)^Kronecker(13, k).
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EXAMPLE
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G.f. = q - q^2 + q^3 - 2*q^4 + q^5 + q^7 + q^8 - q^9 - 2*q^10 + 2*q^13 + 2*q^14 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 13, k], {k, n - 1}], {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 - x^k)^kronecker( 13, k), 1 + x * O(x^n)), n))}; /* Michael Somos, Oct 24 2005 */
(PARI) {a(n) = my(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^2 - v + u*v^3 + u^3*v^2 + 2*u*v * (1 - u + v + u*v), k+1) / 2); polcoeff(A, 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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