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%I #22 Jan 05 2024 12:29:14
%S 1,0,1,0,-1,1,0,-1,-2,1,0,0,-1,-3,1,0,0,2,0,-4,1,0,1,1,5,2,-5,1,0,0,2,
%T 0,8,5,-6,1,0,1,-2,0,-5,10,9,-7,1,0,0,0,-7,-4,-15,10,14,-8,1,0,0,-2,0,
%U -10,-6,-30,7,20,-9,1,0,0,-2,0,8,-5,0,-49,0,27,-10,1
%N A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.
%C The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.
%H M. Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
%H S. R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H V. Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H M. Newman, <a href="/A000727/a000727.pdf">A table of the coefficients of the powers of eta(tau)</a>, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
%H Tim Silverman, <a href="http://arxiv.org/abs/1612.08085">Counting Cliques in Finite Distant Graphs</a>, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%e [ 0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
%e [ 1] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, ... A010815
%e [ 2] 1, -2, -1, 2, 1, 2, -2, 0, -2, -2, ... A002107
%e [ 3] 1, -3, 0, 5, 0, 0, -7, 0, 0, 0, ... A010816
%e [ 4] 1, -4, 2, 8, -5, -4, -10, 8, 9, 0, ... A000727
%e [ 5] 1, -5, 5, 10, -15, -6, -5, 25, 15, -20, ... A000728
%e [ 6] 1, -6, 9, 10, -30, 0, 11, 42, 0, -70, ... A000729
%e [ 7] 1, -7, 14, 7, -49, 21, 35, 41, -49, -133, ... A000730
%e [ 8] 1, -8, 20, 0, -70, 64, 56, 0, -125, -160, ... A000731
%e [ 9] 1, -9, 27, -12, -90, 135, 54, -99, -189, -85, ... A010817
%e [10] 1, -10, 35, -30, -105, 238, 0, -260, -165, 140, ... A010818
%e A001489, v , A167541, v , A319931, v , diagonal: A008705
%e A080956 A319930 A319932
%p DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
%p A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
%p seq(coeff(%, x, j), j=0..len-1) end:
%p seq(print([n], A319933row(n, 10)), n=0..10);
%t eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
%t A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
%t Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 10 2018 *)
%o (Sage)
%o from sage.modular.etaproducts import qexp_eta
%o def A319933row(n, len):
%o return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
%o for n in (0..10):
%o print(A319933row(n, 10))
%o (Julia) # DedekindEta is defined in A000594
%o for n in 0:10
%o DedekindEta(10, n) |> println
%o end
%Y Transpose of A286354.
%Y Cf. A078521, A319574 (JacobiTheta3).
%K sign,tabl
%O 0,9
%A _Peter Luschny_, Oct 02 2018
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