This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A039924 #63 Feb 16 2020 00:48:45
%S A039924 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,
%T A039924 -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,
%U A039924 -21,-31,-27,-37,-31,-40,-33,-41,-31,-39,-27,-33,-18,-24,-6,-11,9,5,26,23
%N A039924 G.f.: Sum_{k>=0} x^(k^2)*(-1)^k/(Product_{i=1..k} 1-x^i).
%C A039924 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
%C A039924 Coefficients in expansion of determinant of infinite tridiagonal matrix shown below in powers of x^2 (Lehmer 1973):
%C A039924 1 x 0 0 0 0 ...
%C A039924 x 1 x^2 0 0 0 ...
%C A039924 0 x^2 1 x^3 0 0 ...
%C A039924 0 0 x^3 1 x^4 0 ...
%C A039924 ... ... ... ... ... ... ...
%C A039924 Convolution inverse of A003116.
%D A039924 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11).
%D A039924 G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, editors, Collected Papers of Srinivasa Ramanujan, Cambridge, 1923; p. 354.
%D A039924 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.
%D A039924 Herman P. Robinson, personal communication to _N. J. A. Sloane_.
%H A039924 Alois P. Heinz, <a href="/A039924/b039924.txt">Table of n, a(n) for n = 0..5000</a> (first 1001 terms from T. D. Noe)
%H A039924 Shalosh B. Ekhad, Doron Zeilberger, <a href="https://arxiv.org/abs/1808.06730">D.H. Lehmer's Tridiagonal determinant: An Etude in (Andrews-Inspired) Experimental Mathematics</a>, arXiv:1808.06730 [math.CO], 2018.
%H A039924 Herman P. Robinson, <a href="/A003116/a003116.pdf">Letter to N. J. A. Sloane, Nov 13 1973</a>.
%F A039924 a(n) = A286041(n) - A286316(n) (conjectured). - _George Beck_, May 05 2017
%F A039924 Proof from _Doron Zeilberger_, Aug 20 2018: (Start)
%F A039924 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)).
%F A039924 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).
%F A039924 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)
%e A039924 G.f. = 1 - x - x^2 - x^3 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + x^12 + ...
%p A039924 qq:=n->mul( 1-(-q)^i, i=1..n); add (q^(n^2)/qq(n),n=0..100): series(t1,q,99);
%t A039924 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 *)
%t A039924 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 *)
%o A039924 (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 */
%Y A039924 Cf. A003116, A224898.
%K A039924 sign,nice
%O A039924 0,9
%A A039924 _N. J. A. Sloane_
%E A039924 More terms from _Vladeta Jovovic_, Mar 05 2001