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A278074 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 = 4.

%I A278074 #22 Mar 24 2020 17:36:00
%S A278074 1,0,1,0,1,70,0,1,990,34650,0,1,16510,2702700,63063000,0,1,261630,
%T A278074 213519150,17459442000,305540235000,0,1,4196350,17651304000,
%U A278074 4350310965000,231905038365000,3246670537110000
%N A278074 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 = 4.
%H A278074 Robert Israel, <a href="/A278074/b278074.txt">Table of n, a(n) for n = 0..8510</a>
%F A278074 E.g.f.: 1/(1-t*((cosh(x)+cos(x))/2-1)), nonzero terms.
%e A278074 Triangle starts:
%e A278074 [1]
%e A278074 [0, 1]
%e A278074 [0, 1,     70]
%e A278074 [0, 1,    990,     34650]
%e A278074 [0, 1,  16510,   2702700,    63063000]
%e A278074 [0, 1, 261630, 213519150, 17459442000, 305540235000]
%p A278074 P := proc(m,n) option remember; if n = 0 then 1 else
%p A278074 add(binomial(m*n,m*k)* P(m,n-k)*x, k=1..n) fi end:
%p A278074 for n from 0 to 6 do PolynomialTools:-CoefficientList(P(4,n), x) od;
%p A278074 # Alternatively:
%p A278074 A278074_row := proc(n) 1/(1-t*((cosh(x)+cos(x))/2-1)); expand(series(%,x,4*n+1));
%p A278074 (4*n)!*coeff(%,x,4*n); PolynomialTools:-CoefficientList(%,t) end:
%p A278074 for n from 0 to 5 do A278074_row(n) od;
%t A278074 With[{m = 4}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 24, m}]];
%t A278074 Function[arg, CoefficientList[arg, t]] /@ % // Flatten
%o A278074 (Sage) # uses [P from A278073]
%o A278074 def A278074_row(n): return list(P(4, n))
%o A278074 for n in (0..6): print(A278074_row(n)) # _Peter Luschny_, Mar 24 2020
%Y A278074 Cf. A014608 (diagonal), A243665 (row sums), A211212 (alternating row sums), A281480 (central coefficients).
%Y A278074 Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3).
%Y A278074 Cf. A327024 (refinement).
%K A278074 nonn,tabl
%O A278074 0,6
%A A278074 _Peter Luschny_, Jan 22 2017