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A278073
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Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.
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18
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1, 0, 1, 0, 1, 20, 0, 1, 168, 1680, 0, 1, 1364, 55440, 369600, 0, 1, 10920, 1561560, 33633600, 168168000, 0, 1, 87380, 42771456, 2385102720, 34306272000, 137225088000, 0, 1, 699048, 1160164320, 158411809920, 5105916816000, 54752810112000, 182509367040000
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f.: 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1)), nonzero terms.
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EXAMPLE
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Triangle begins:
[1]
[0, 1]
[0, 1, 20]
[0, 1, 168, 1680]
[0, 1, 1364, 55440, 369600]
[0, 1, 10920, 1561560, 33633600, 168168000]
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MAPLE
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P := proc(m, n) option remember; if n = 0 then 1 else
add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3, n), x) od;
# Alternatively:
1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));
expand(series(%, x, 3*n+1)); (3*n)!*coeff(%, x, 3*n);
PolynomialTools:-CoefficientList(%, t) end:
for n from 0 to 6 do A278073_row(n) od;
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MATHEMATICA
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With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten
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PROG
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(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
@cached_function
def P(m, n):
if n == 0: return R(1)
return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
def A278073_row(n): return list(P(3, 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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