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A291452 Triangle read by rows, expansion of e.g.f. exp(x*(cos(z) + cosh(z) - 2)/2), nonzero coefficients of z.

%I A291452 #12 Jul 23 2019 04:23:52
%S A291452 1,0,1,0,1,35,0,1,495,5775,0,1,8255,450450,2627625,0,1,130815,
%T A291452 35586525,727476750,2546168625,0,1,2098175,2941884000,181262956875,
%U A291452 1932541986375,4509264634875
%N A291452 Triangle read by rows, expansion of e.g.f. exp(x*(cos(z) + cosh(z) - 2)/2), nonzero coefficients of z.
%e A291452 Triangle starts:
%e A291452 [1]
%e A291452 [0, 1]
%e A291452 [0, 1,      35]
%e A291452 [0, 1,     495,       5775]
%e A291452 [0, 1,    8255,     450450,      2627625]
%e A291452 [0, 1,  130815,   35586525,    727476750,    2546168625]
%e A291452 [0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875]
%p A291452 CL := (f,x) -> PolynomialTools:-CoefficientList(f,x):
%p A291452 A291452_row := proc(n) exp(x*(cos(z)+cosh(z)-2)/2):
%p A291452 series(%, z, 88): CL((4*n)!*coeff(series(%,z,4*(n+1)),z,4*n),x) end:
%p A291452 for n from 0 to 7 do A291452_row(n) od;
%p A291452 # Alternative:
%p A291452 A291452row := proc(n) local P; P := proc(m, n) option remember;
%p A291452 if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
%p A291452 CL(P(4, n), x); seq(%[k+1]/k!, k=0..n) end: # _Peter Luschny_, Sep 03 2018
%t A291452 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 A291452 row[n_] := Module[{cl = CoefficientList[P[4, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
%t A291452 Table[row[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Jul 23 2019, after _Peter Luschny_ *)
%Y A291452 Cf. A048993 (m=1), A156289 (m=2), A291451 (m=3), this seq. (m=4).
%Y A291452 Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).
%Y A291452 Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).
%Y A291452 Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).
%K A291452 nonn,tabl
%O A291452 0,6
%A A291452 _Peter Luschny_, Sep 07 2017