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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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%I #63 Feb 16 2020 00:48:45

%S 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 -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 -21,-31,-27,-37,-31,-40,-33,-41,-31,-39,-27,-33,-18,-24,-6,-11,9,5,26,23

%N G.f.: Sum_{k>=0} x^(k^2)*(-1)^k/(Product_{i=1..k} 1-x^i).

%C 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 Coefficients in expansion of determinant of infinite tridiagonal matrix shown below in powers of x^2 (Lehmer 1973):

%C 1 x 0 0 0 0 ...

%C x 1 x^2 0 0 0 ...

%C 0 x^2 1 x^3 0 0 ...

%C 0 0 x^3 1 x^4 0 ...

%C ... ... ... ... ... ... ...

%C Convolution inverse of A003116.

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11).

%D G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, editors, Collected Papers of Srinivasa Ramanujan, Cambridge, 1923; p. 354.

%D 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 Herman P. Robinson, personal communication to _N. J. A. Sloane_.

%H 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 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 Herman P. Robinson, <a href="/A003116/a003116.pdf">Letter to N. J. A. Sloane, Nov 13 1973</a>.

%F a(n) = A286041(n) - A286316(n) (conjectured). - _George Beck_, May 05 2017

%F Proof from _Doron Zeilberger_, Aug 20 2018: (Start)

%F 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 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 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 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 qq:=n->mul( 1-(-q)^i, i=1..n); add (q^(n^2)/qq(n),n=0..100): series(t1,q,99);

%t 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 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 (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 Cf. A003116, A224898.

%K sign,nice

%O 0,9

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Mar 05 2001