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 A299018 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial P(n) = n*(x + 1)*P(n - 1) - (n - 2)^2*x*P(n - 2). 0
 1, 2, 2, 6, 11, 6, 24, 60, 60, 24, 120, 366, 501, 366, 120, 720, 2532, 4242, 4242, 2532, 720, 5040, 19764, 38268, 46863, 38268, 19764, 5040, 40320, 172512, 373104, 528336, 528336, 373104, 172512, 40320, 362880, 1668528, 3942108, 6237828, 7213761, 6237828, 3942108, 1668528, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA P(0) = 0, P(1) = 1 and P(n) = n * (x + 1) * P(n - 1) - (n - 2)^2 * x * P(n - 2). EXAMPLE For n = 3, the polynomial is 6*x^2 + 11*x + 6. The first few polynomials, as a table: [1], [2,    2], [6,    11,    6], [24,   60,    60,  24], [120,  366,   501, 366, 120] MAPLE P:= proc(n) option remember; expand(`if`(n<2, n,       n*(x+1)*P(n-1)-(n-2)^2*x*P(n-2)))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(P(n)): seq(T(n), n=1..12);  # Alois P. Heinz, Jan 31 2018 A := proc(n, k) ## n >= 0 and k = 0 .. n     option remember;     if n = 0 and k = 0 then         1     elif n > 0 and k >= 0 and k <= n then         (n+1)*(A(n-1, k)+A(n-1, k-1))-(n-1)^2*A(n-2, k-1)     else         0     end if; end proc: # Yu-Sheng Chang, Apr 14 2020 MATHEMATICA P[n_] := P[n] = Expand[If[n < 2, n, n (x+1) P[n-1] - (n-2)^2 x P[n-2]]]; row[n_] := CoefficientList[P[n], x]; row /@ Range[12] // Flatten (* Jean-François Alcover, Dec 10 2019 *) PROG (Sage) @cached_function def poly(n):     x = polygen(ZZ, 'x')     if n < 1:         return x.parent().zero()     elif n == 1:         return x.parent().one()     else:         return n * (x + 1) * poly(n - 1) - (n - 2)**2 * x * poly(n - 2) CROSSREFS Very similar to A298854. Row sums are A277382(n-1) for n>0. Leftmost and rightmost columns are A000142. Alternating row sums are A177145. Alternating row sum of row 2*n+1 is A001818(n). Sequence in context: A036052 A308260 A279212 * A269830 A275312 A209026 Adjacent sequences:  A299015 A299016 A299017 * A299019 A299020 A299021 KEYWORD tabl,nonn,easy,changed AUTHOR F. Chapoton, Jan 31 2018 STATUS approved

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)