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%I #14 Apr 12 2023 10:27:28
%S 1,1,0,2,0,0,6,0,1,0,24,0,8,0,0,120,0,60,0,1,0,720,0,480,0,32,0,0,
%T 5040,0,4200,0,546,0,1,0,40320,0,40320,0,8064,0,128,0,0,362880,0,
%U 423360,0,115920,0,4920,0,1,0,3628800,0,4838400,0,1693440,0,130560,0,512,0,0
%N Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).
%C See A196776 for a row reversed form of this triangle. - _Peter Bala_, Oct 06 2011
%F P(n, 0) = A000142(n).
%F P(n, 1) = A006154(n).
%F P(n, 2) = A191277(n).
%F P(n, i) = A000111(n+1), where i is the imaginary unit.
%F P(n, i)*2^n = A000828(n+1).
%F P(n, 1/2)*2^n = A000557(n).
%F P(n, 1/3)*3^n = A107403(n).
%F P(n, i/2)*2^n = A007289(n).
%F G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
%F GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
%F [x^2] P(n+1, x) = A005990(n).
%e The sequence of polynomials P(n, x) begins:
%e [0] 1;
%e [1] 1;
%e [2] 2;
%e [3] 6 + x^2;
%e [4] 24 + 8*x^2;
%e [5] 120 + 60*x^2 + x^4;
%e [6] 720 + 480*x^2 + 32*x^4;
%e [7] 5040 + 4200*x^2 + 546*x^4 + x^6.
%p A193474_polynom := proc(n,x) local k, j;
%p add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);
%t p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 27 2014 *)
%Y Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.
%K nonn,tabl
%O 1,4
%A Peter Luschny, Aug 01 2011
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