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A039924
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G.f.: Sum_{k>=0} x^(k^2)*(-1)^k/(Product_{i=1..k} 1-x^i).
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9
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1, -1, -1, -1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, -2, -1, -3, -3, -4, -3, -5, -3, -4, -2, -3, 0, -1, 3, 2, 5, 5, 9, 7, 11, 9, 13, 10, 13, 9, 12, 7, 9, 3, 5, -3, -1, -9, -7, -17, -15, -24, -21, -31, -27, -37, -31, -40, -33, -41, -31, -39, -27, -33, -18, -24, -6, -11, 9, 5, 26, 23
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OFFSET
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0,9
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COMMENTS
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Ramanujan used the form Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1-(-x)^i), which is obtained by changing the sign of x. - Michael Somos, Jul 20 2003
Coefficients in expansion of determinant of infinite tridiagonal matrix shown below in powers of x^2 (Lehmer 1973):
1 x 0 0 0 0 ...
x 1 x^2 0 0 0 ...
0 x^2 1 x^3 0 0 ...
0 0 x^3 1 x^4 0 ...
... ... ... ... ... ... ...
Convolution inverse of A003116.
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11).
G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, editors, Collected Papers of Srinivasa Ramanujan, Cambridge, 1923; p. 354.
D. H. Lehmer, Combinatorial and cyclotomic properties of certain tridiagonal matrices. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 53-74. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. MR0441852.
Herman P. Robinson, personal communication to N. J. A. Sloane.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from T. D. Noe)
Shalosh B. Ekhad, Doron Zeilberger, D.H. Lehmer's Tridiagonal determinant: An Etude in (Andrews-Inspired) Experimental Mathematics, arXiv:1808.06730 [math.CO], 2018.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
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FORMULA
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a(n) = A286041(n) - A286316(n) (conjectured). - George Beck, May 05 2017
Proof from Doron Zeilberger, Aug 20 2018: (Start)
The generating function for partitions whose parts differ by at least 2 with exactly k parts is (famously) q^(k^2)/((1-q)*...*(1-q^k)).
Indeed, if you take any such partition and remove 1 from the smallest part, 3 from the second-smallest part, etc., you remove 1+3+...+(2k-1) = k^2 and are left with an ordinary partition whose number of parts is <= k whose generating function is 1/((1-q)*...*(1-q)^k).
Summing these up famously gives the generating function for partitions whose differences is >= 2. Sticking a (-1)^k in front gives the generating function for the difference between such partitions with an even number of parts and an odd number of parts, since (-1)^even=1 and (-1)^odd=-1. (End)
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EXAMPLE
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G.f. = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 + ...
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MAPLE
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qq:=n->mul( 1-(-q)^i, i=1..n); add (q^(n^2)/qq(n), n=0..100): series(t1, q, 99);
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MATHEMATICA
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CoefficientList[ Series[ Sum[x^k^2*(-1)^k / Product[1-x^i, {i, 1, k}], {k, 0, 100}], {x, 0, 100}], x][[1 ;; 72]] (* Jean-François Alcover, Apr 08 2011 *)
a[ n_] := If[n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^k^2 / QPochhammer[ x, x, k], {k, 0, Sqrt[n]}], {x, 0, n}]] (* Michael Somos, Jan 04 2014 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, x^i - 1, 1 + x * O(x^n))), n))} /* Michael Somos, Jul 20 2003 */
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CROSSREFS
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Cf. A003116, A224898.
Sequence in context: A093654 A342627 A220115 * A349218 A275346 A037888
Adjacent sequences: A039921 A039922 A039923 * A039925 A039926 A039927
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KEYWORD
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sign,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Vladeta Jovovic, Mar 05 2001
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STATUS
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approved
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