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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)*C(k,j)*(k-2*j)^n*x^(n-k). 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

P(n,0) = A000142(n)

P(n,1) = A006154(n)

P(n,2) = A191277(n)

P(n,I) = A000111(n+1)

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).

coeff_[x^2] P(n+1,x) = A005990(n)

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

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

[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

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 *)

CROSSREFS

Cf. A196776.

Sequence in context: A035536 A205974 A098643 * A241020 A277444 A274710

Adjacent sequences:  A193471 A193472 A193473 * A193475 A193476 A193477

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Aug 01 2011

STATUS

approved

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Last modified June 20 22:19 EDT 2018. Contains 305615 sequences. (Running on oeis4.)