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

Author

Peter Luschny, Aug 01 2011

Keywords

• nonn
• tabl

Comments

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

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

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

Programs

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);

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).