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%I #52 Jun 25 2019 09:48:31
%S 1,6,15,26,45,66,82,120,156,170,231,276,290,390,435,438,561,630,651,
%T 780,861,842,1020,1170,1095,1326,1431,1370,1716,1740,1682,2016,2145,
%U 2132,2415,2550,2353,2850,3120,2810,3321,3486,3285,3906,4005,3722,4350
%N Expansion of Jacobi theta constant theta_2^6 /(64q^(3/2)).
%C Number of representations of n as sum of 6 triangular numbers. - _Michel Marcus_, Oct 24 2012. See the Ono et al. link.
%D B. C. Berndt, Fragments by Ramanujan on Lambert series, in Number Theory and Its Applications, K. Gyory and S. Kanemitsu, eds., Kluwer, Dordrecht, 1999, pp. 35-49, see Entry 6.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%H Seiichi Manyama, <a href="/A008440/b008440.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H B. C. Berndt, <a href="http://www.math.uiuc.edu/~berndt/lambertseries.ps">Fragments by Ramanujan on Lambert series</a>.
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 4.
%F Expansion of Ramanujan phi^6(q) in powers of q.
%F Expansion of q^(-3/4)(eta(q^2)^2/eta(q))^6 in powers of q.
%F Euler transform of period 2 sequence [6, -6, ...]. - _Michael Somos_, May 23 2006
%F G.f.: (Sum_{n>=0} x^((n^2+n)/2))^6.
%F a(n) = (-1/8)*Sum_{d divides (4n+3)} Chi_2(4;d)*d^2. - _Michel Marcus_, Oct 24 2012. See the Ono et al. link. Theorem 4.
%F a(n) =(-1/8)*A002173(4*n+3). This is the preceding formula. - _Wolfdieter Lang_, Jan 12 2017
%F a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 06 2017
%F G.f.: exp(Sum_{k>=1} 6*(x^k/k)/(1 + x^k)). - _Ilya Gutkovskiy_, Jul 31 2017
%e G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 45*x^4 + 66*x^5 + 82*x^6 + ... - _Michael Somos_, Jun 25 2019
%e G.f. = q^3 + 6*q^7 + 15*q^11 + 26*q^15 + 45*q^19 + 66*q^23 + 82*q^27 + ...
%t CoefficientList[(QPochhammer[q^2]^2 / QPochhammer[q])^6 + O[q]^50, q] (* _Jean-François Alcover_, Nov 05 2015 *)
%t a[ n_] := If[ n < 0, 0, -DivisorSum[ 4 n + 3, Re[I^(# - 1)] #^2 &] / 8]; (* _Michael Somos_, Jun 25 2019 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x * O(x^n))^6, n))}; /* _Michael Somos_, May 23 2006 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^6, n))}; /* _Michael Somos_, May 23 2006 */
%o (PARI) {a(n)= -sumdiv(4*n + 3, d, real(I^(d-1))*d^2)/8}; /* _Michael Somos_, Oct 24 2012 */
%Y Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809, A002173.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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