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 A128966 Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2]. 5
 0, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 20, 20, 8, 1, 1, 10, 34, 50, 34, 10, 1, 1, 12, 52, 104, 104, 52, 12, 1, 1, 14, 74, 190, 258, 190, 74, 14, 1, 1, 16, 100, 316, 552, 552, 316, 100, 16, 1, 1, 18, 130, 490, 1058, 1362, 1058, 490, 130, 18, 1, 1, 20, 164 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS A variant of A008288 (they satisfy the same recurrence). LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened FORMULA P[n](x) = (x+1) * ( ((x+1+sqrt(x^2+6x+1))/2)^n - ((x+1-sqrt(x^2+6x+1))/2)^n ) / sqrt(x^2+6x+1) - Max Alekseyev, Mar 10 2008 P[n](x) = (x+1) * (sqrt(x)*I)^(n-1) * U[n-1](-I*(x+1)/sqrt(x)/2), where U[n](t) is Chebyshev polynomial of the 2nd kind. - Max Alekseyev, Mar 10 2008 EXAMPLE Triangle begins: 0 1, 1 1, 2, 1 1, 4, 4, 1 1, 6, 10, 6, 1 1, 8, 20, 20, 8, 1 1, 10, 34, 50, 34, 10, 1 1, 12, 52, 104, 104, 52, 12, 1 1, 14, 74, 190, 258, 190, 74, 14, 1 1, 16, 100, 316, 552, 552, 316, 100, 16, 1 MAPLE P[0]:=0; P[1]:=x+1; for n from 2 to 14 do P[n]:=expand((x+1)*P[n-1]+x*P[n-2]); lprint(P[n]); lprint(seriestolist(series(P[n], x, 200))); od: MATHEMATICA t[n_, k_] := 2^(1-n)*Binomial[n, k]*Sum[Binomial[n, 2*m+1]*HypergeometricPFQ[{-k, -m, k-n}, {1/2-n/2, -n/2}, -1], {m, 0, (n-1)/2}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014, after Max Alekseyev *) PROG (PARI) { T(n, k) = sum(m=0, (n-1)\2, binomial(n, 2*m+1) * sum(j=0, m, binomial(m, j) * binomial(n-2*j, k-j) * 2^(2*j+1-n) ) ) } - Max Alekseyev, Mar 10 2008 (Haskell) a128966 n k = a128966_tabl !! n !! k a128966_row n = a128966_tabl !! n a128966_tabl = map fst \$ iterate    (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) \$                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([0], [1, 1]) -- Reinhard Zumkeller, Jul 20 2013 CROSSREFS Cf. A163271 (row sums), A110170 (central terms). Cf. A102413. Sequence in context: A156580 A157528 A132731 * A055907 A259698 A274643 Adjacent sequences:  A128963 A128964 A128965 * A128967 A128968 A128969 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, May 10 2007 STATUS approved

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Last modified May 26 09:37 EDT 2020. Contains 334620 sequences. (Running on oeis4.)