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%I #30 Mar 11 2021 03:17:51
%S 1,1,2,3,4,6,8,11,15,20,26,34,44,56,71,90,112,140,174,214,263,322,392,
%T 476,576,694,834,1000,1194,1423,1692,2005,2372,2800,3296,3874,4544,
%U 5318,6214,7248,8438,9808,11383,13188,15258,17628,20334,23426,26952,30966,35536,40730
%N Number of partitions of n into parts that are odd or == +- 2 (mod 10).
%C Andrews gives a second interpretation of these numbers and refers to them as a cousin of the Rogers-Ramanujan numbers.
%C Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
%C Also number of partitions of n into parts not congruent to 0 or +-4 (mod 10). [Agarwal-Andrews (1987).] - _N. J. A. Sloane_, Nov 30 2019
%D Agarwal, A. K., and George E. Andrews. "Rogers-Ramanujan identities for partitions with 'N copies of N'." Journal of Combinatorial Theory, Series A 45.1 (1987): 40-49. See Theorem 1.
%H Vincenzo Librandi, <a href="/A133153/b133153.txt">Table of n, a(n) for n = 0..1000</a>
%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/cbms/066">q-series</a>, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 7, Eq. (1.3). MR0858826 (88b:11063).
%H G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 10.)
%H Mircea Merca, <a href="http://cdm.ucalgary.ca/cdm/index.php/cdm/article/view/757">On the number of partitions into odd parts or congruent to +/-2 mod 10</a>, Contributions to Discrete Mathematics, Vol. 13, No. 1 (2018).
%F Expansion of f(-q^4, -q^6) / f(-q) in powers of q where f() is Ramanujan's theta function.
%F Euler transform of period 10 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...]. - _Michael Somos_, Sep 30 2007
%F G.f.: (Sum_{k>=0} x^(2*k^2) / ( (1 - x^2) * (1 - x^4) * ... * (1 - x^(2*k)) ) )/ Product_{k>0} 1 - x^(2*k-1).
%F G.f.: Sum_{k>=0} x^(k*(3*k+1)/2) * (1 + x) * (1 + x^2) * ... * (1 + x^(2*k)) / ( (1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1)) ).
%e G.f. = 1 + q + 2*q^2 + 3*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 11*q^7 + 15*q^8 + ...
%p lis1:=[1,2,3,5,7,8,9];
%p g1:= k -> 1/mul(1-x^(10*k+i),i in lis1);
%p g2:=mul(g1(k),k=0..100);
%p series(g2,x,50); # _N. J. A. Sloane_, Nov 30 2019
%t th[m_][a_, b_] := Sum[a^(n*((n+1)/2))*b^(n*((n-1)/2)), {n, -m, m}]; th[m_][-q_] := th[m][-q, -q^2];
%t f[s_List] := (m++; CoefficientList[ Series[ th[m][-q^4, -q^6]/th[m][-q], {q, 0, 51}], q]); FixedPoint[f, m = 0; {}] (* _Jean-François Alcover_, Jul 19 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^4, x^10] QPochhammer[ x^6, x^10] QPochhammer[ x^10] / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Mar 27 2014 *)
%o (PARI) {a(n) = local(t); if( n<0, 0, t = 1 / (1 - x) + x * O(x^n); polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1)\6, t = t * x^(3*k - 1) / (1 - x^k) / (1 - x^(2*k + 1)) + x * O(x^n), t), n))}; /* _Michael Somos_, Sep 30 2007 */
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Sep 22 2007
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