A193474 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).
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, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0
Offset: 1
Examples
The sequence of polynomials P(n, x) begins: [0] 1; [1] 1; [2] 2; [3] 6 + x^2; [4] 24 + 8*x^2; [5] 120 + 60*x^2 + x^4; [6] 720 + 480*x^2 + 32*x^4; [7] 5040 + 4200*x^2 + 546*x^4 + x^6.
Crossrefs
Programs
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Maple
A193474_polynom := proc(n,x) local k, j; 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);
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Mathematica
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 *)
Formula
P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).
Comments