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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 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 October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)