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A278073 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.

%I A278073 #20 Mar 24 2020 08:40:06
%S A278073 1,0,1,0,1,20,0,1,168,1680,0,1,1364,55440,369600,0,1,10920,1561560,
%T A278073 33633600,168168000,0,1,87380,42771456,2385102720,34306272000,
%U A278073 137225088000,0,1,699048,1160164320,158411809920,5105916816000,54752810112000,182509367040000
%N A278073 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.
%F A278073 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.
%e A278073 Triangle begins:
%e A278073 [1]
%e A278073 [0, 1]
%e A278073 [0, 1,    20]
%e A278073 [0, 1,   168,    1680]
%e A278073 [0, 1,  1364,   55440,   369600]
%e A278073 [0, 1, 10920, 1561560, 33633600, 168168000]
%p A278073 P := proc(m, n) option remember; if n = 0 then 1 else
%p A278073 add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
%p A278073 for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3,n), x) od;
%p A278073 # Alternatively:
%p A278073 A278073_row := proc(n)
%p A278073 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));
%p A278073 expand(series(%,x,3*n+1)); (3*n)!*coeff(%,x,3*n);
%p A278073 PolynomialTools:-CoefficientList(%,t) end:
%p A278073 for n from 0 to 6 do A278073_row(n) od;
%t A278073 With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];
%t A278073 Function[arg, CoefficientList[arg, t]] /@ % // Flatten
%o A278073 (Sage)
%o A278073 R = PowerSeriesRing(ZZ, 'x')
%o A278073 x = R.gen().O(30)
%o A278073 @cached_function
%o A278073 def P(m, n):
%o A278073     if n == 0: return R(1)
%o A278073     return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
%o A278073 def A278073_row(n): return list(P(3, n))
%o A278073 for n in (0..6): print(A278073_row(n)) # _Peter Luschny_, Mar 24 2020
%Y A278073 Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).
%Y A278073 Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).
%K A278073 nonn,tabl
%O A278073 0,6
%A A278073 _Peter Luschny_, Jan 22 2017