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Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.
2

%I #19 Mar 12 2013 12:45:25

%S 1,-23,3592,-419,1,-94,169659,-65838,1092873176,145023,1,-213,1312544,

%T -723721,44648582886,9188934683,166629520876208,2791651635293,1,-475,

%U 9032603,-9455070,3949512899743,-97215753021,9776785708507683,-53144327916296,-134884469547631

%N Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

%C For an algorithm to compute the partition class polynomial Hpart_n(x) see the Bruinier-Ono-Sutherland paper, 3.3. Algorithm 3, p. 15-19.

%C Note that the absolute value of T(n,2) is also the trace Tr(n) = A183011(n), the numerator of the finite algebraic formula for the number of partitions of n. The formula is p(n) = Tr(n)/(24*n - 1). See theorem 1.1 in the Bruinier-Ono paper.

%H J. H. Bruinier and K. Ono, <a href="http://www.aimath.org/news/partition/brunier-ono.pdf">Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms</a>

%H J. H. Bruinier, K. Ono, A. V. Sutherland, <a href="http://arxiv.org/abs/1301.5672">Class polynomials for nonholomorphic modular functions</a>

%H A. V. Sutherland, <a href="http://math.mit.edu/~drew/Pfiles/">Partition class polynomials</a>, Hpart_n(x), for n = 1..770

%F abs(T(n,2))/(24n-1) = A183011(n)/A183010(n) = A000041(n).

%e For n = 1 the first partition class polynomial Hpart_1(x) is x^3 - 23*x^2 + 3592/23*x - 419, so the numerators of the coefficients are 1, -23, 3592, -419.

%e Triangle begins:

%e 1, -23, 3592, -419;

%e 1, -94, 169659, -65838, 1092873176, 145023;

%e 1, -213, 1312544, -723721, 44648582886, 9188934683, 166629520876208, 2791651635293;

%e 1, -475, 9032603, -9455070, 3949512899743, -97215753021, 9776785708507683, -53144327916296, -134884469547631;

%e ...

%Y Row n has length 1 + A188569(n). Absolute values of column 2 give A183011. Columns 3-4: A183007, A187218. For denominators see A222032.

%Y Cf. A000041, A183010, A220515.

%K sign,frac,tabf

%O 1,2

%A _Omar E. Pol_, Mar 04 2013