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A319933
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A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.
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4
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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, 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, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1
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OFFSET
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0,9
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COMMENTS
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The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.
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LINKS
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EXAMPLE
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[ 0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
[ 1] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, ... A010815
[ 2] 1, -2, -1, 2, 1, 2, -2, 0, -2, -2, ... A002107
[ 3] 1, -3, 0, 5, 0, 0, -7, 0, 0, 0, ... A010816
[ 4] 1, -4, 2, 8, -5, -4, -10, 8, 9, 0, ... A000727
[ 5] 1, -5, 5, 10, -15, -6, -5, 25, 15, -20, ... A000728
[ 6] 1, -6, 9, 10, -30, 0, 11, 42, 0, -70, ... A000729
[ 7] 1, -7, 14, 7, -49, 21, 35, 41, -49, -133, ... A000730
[ 8] 1, -8, 20, 0, -70, 64, 56, 0, -125, -160, ... A000731
[ 9] 1, -9, 27, -12, -90, 135, 54, -99, -189, -85, ... A010817
[10] 1, -10, 35, -30, -105, 238, 0, -260, -165, 140, ... A010818
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MAPLE
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DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);
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MATHEMATICA
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eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
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PROG
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(Sage)
from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
print(A319933row(n, 10))
(Julia) # DedekindEta is defined in A000594
for n in 0:10
DedekindEta(10, n) |> println
end
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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