A039924 G.f.: Sum_{k>=0} x^(k^2)*(-1)^k/(Product_{i=1..k} 1-x^i).
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
Offset: 0
Examples
G.f. = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 + ...
References
- 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.
Links
- 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.
Programs
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Maple
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
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 *) -
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 */
Formula
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)
Extensions
More terms from Vladeta Jovovic, Mar 05 2001
Comments