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A080995
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Characteristic function of generalized pentagonal numbers A001318.
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19
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1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A090864(n))=0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
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REFERENCES
| Cooper, S. and Hirschhorn, M. D., Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 81, Article 331.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1001
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for characteristic functions
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FORMULA
| Expansion of phi(-q^3) / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 14 2007
Expansion of psi(q) - q * psi(q^9) in powers of q^3 where psi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.
a(n) = b(24*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p>3. - Michael Somos, Jun 06 2005
Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A089810.
G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos, Jan 26 2008
G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].
a(n) = |A010815(n)| = A089806(2*n) = A033683(24*n + 1).
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EXAMPLE
| 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + ...
q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> x^(1/2)}, {x, 0, n}]] (* Michael Somos, Nov 18 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))}
(PARI) {a(n) = issquare(24*n + 1)} /* Michael Somos, Apr 13 2005 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) *eta(x^6 + A)), n))}
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CROSSREFS
| Cf. A001318 (support), A010815 (absolute values), A033683, A089806.
Sequence in context: A121373 A133985 A143062 A199918 A074910 A115356 A115359
Adjacent sequences: A080992 A080993 A080994 * A080996 A080997 A080998
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Michael Somos, Feb 27, 2003
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EXTENSIONS
| Minor edits by N. J. A. Sloane, Feb 03 2012
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