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A319083
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Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.
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5
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1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} sigma(n-k)*p(k, x).
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EXAMPLE
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Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 3, 1;
[3] 0, 4, 6, 1;
[4] 0, 7, 17, 9, 1;
[5] 0, 6, 38, 39, 12, 1;
[6] 0, 12, 70, 120, 70, 15, 1;
[7] 0, 8, 116, 300, 280, 110, 18, 1;
[8] 0, 15, 185, 645, 885, 545, 159, 21, 1;
[9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;
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MAPLE
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P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-sigma(n-k)*P(k, x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
# Uses function PMatrix from A357368.
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
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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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