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A005927
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Theta series of diamond with respect to deep hole.
(Formerly M3262)
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6
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0, 0, 0, 4, 6, 0, 0, 0, 0, 0, 0, 12, 8, 0, 0, 0, 0, 0, 0, 12, 24, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 24, 30, 0, 0, 0, 0, 0, 0, 12, 24, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 12, 48, 0, 0, 0, 0, 0, 0, 28, 24, 0, 0, 0, 0, 0, 0, 36, 48, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0
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OFFSET
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0,4
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COMMENTS
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Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
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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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FORMULA
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Expansion of 4 * q^3 * psi^3(q^8) + (phi^3(q^4) - phi^3(-q^4)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 17 2009
a(8*n + 0) = a(8*n + 1) = a(8*n + 2) = a(8*n + 5) = a(8*n + 6) = a(8*n + 7) = 0. - Michael Somos, Aug 17 2009
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EXAMPLE
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4*q^3 + 6*q^4 + 12*q^11 + 8*q^12 + 12*q^19 + 24*q^20 + 16*q^27 + ... - Michael Somos, Aug 17 2009
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MATHEMATICA
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a[n_]:= SeriesCoefficient[4*q^3*QPochhammer[-q^8, q^8]^3* QPochhammer[q^16, q^16]^3 + (EllipticTheta[3, 0, q^4]^3 - EllipticTheta[3, 0, -q^4]^3)/2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 01 2018 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, if( n%8 == 3, n \= 8; polcoeff( 4 * sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n), if( n%8 == 4, n /= 4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n))^3, n), 0 )))} /* Michael Somos, Aug 17 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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