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A291451 Triangle read by rows, expansion of e.g.f. exp(x*(exp(z)/3 + 2*exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.

%I A291451 #14 Jul 23 2019 04:26:49
%S A291451 1,0,1,0,1,10,0,1,84,280,0,1,682,9240,15400,0,1,5460,260260,1401400,
%T A291451 1401400,0,1,43690,7128576,99379280,285885600,190590400,0,1,349524,
%U A291451 193360720,6600492080,42549306800,76045569600,36212176000
%N A291451 Triangle read by rows, expansion of e.g.f. exp(x*(exp(z)/3 + 2*exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.
%e A291451 Triangle starts:
%e A291451 [1]
%e A291451 [0, 1]
%e A291451 [0, 1,    10]
%e A291451 [0, 1,    84,     280]
%e A291451 [0, 1,   682,    9240,    15400]
%e A291451 [0, 1,  5460,  260260,  1401400,   1401400]
%e A291451 [0, 1, 43690, 7128576, 99379280, 285885600, 190590400]
%p A291451 CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
%p A291451 A291451_row := proc(n) exp(x*(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
%p A291451 series(%, z, 66): CL((3*n)!*coeff(series(%,z,3*(n+1)),z,3*n),x) end:
%p A291451 for n from 0 to 7 do A291451_row(n) od;
%p A291451 # Alternative:
%p A291451 A291451row := proc(n) local P; P := proc(m, n) option remember;
%p A291451 if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
%p A291451 CL(P(3, n), x); seq(%[k+1]/k!, k=0..n) end: # _Peter Luschny_, Sep 03 2018
%t A291451 P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
%t A291451 row[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
%t A291451 Table[row[n], {n, 0, 7}] // Flatten (* _Jean-François Alcover_, Jul 23 2019, after _Peter Luschny_ *)
%Y A291451 Cf. A048993 (m=1), A156289 (m=2), this seq. (m=3), A291452 (m=4).
%Y A291451 Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).
%Y A291451 Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).
%Y A291451 Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).
%K A291451 nonn,tabl
%O A291451 0,6
%A A291451 _Peter Luschny_, Sep 07 2017