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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.
18

%I #20 Mar 24 2020 08:40:06

%S 1,0,1,0,1,20,0,1,168,1680,0,1,1364,55440,369600,0,1,10920,1561560,

%T 33633600,168168000,0,1,87380,42771456,2385102720,34306272000,

%U 137225088000,0,1,699048,1160164320,158411809920,5105916816000,54752810112000,182509367040000

%N 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 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 Triangle begins:

%e [1]

%e [0, 1]

%e [0, 1, 20]

%e [0, 1, 168, 1680]

%e [0, 1, 1364, 55440, 369600]

%e [0, 1, 10920, 1561560, 33633600, 168168000]

%p P := proc(m, n) option remember; if n = 0 then 1 else

%p add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:

%p for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3,n), x) od;

%p # Alternatively:

%p A278073_row := proc(n)

%p 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));

%p expand(series(%,x,3*n+1)); (3*n)!*coeff(%,x,3*n);

%p PolynomialTools:-CoefficientList(%,t) end:

%p for n from 0 to 6 do A278073_row(n) od;

%t With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];

%t Function[arg, CoefficientList[arg, t]] /@ % // Flatten

%o (Sage)

%o R = PowerSeriesRing(ZZ, 'x')

%o x = R.gen().O(30)

%o @cached_function

%o def P(m, n):

%o if n == 0: return R(1)

%o return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))

%o def A278073_row(n): return list(P(3, n))

%o for n in (0..6): print(A278073_row(n)) # _Peter Luschny_, Mar 24 2020

%Y Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).

%Y Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, Jan 22 2017