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A077070 Triangle read by rows: T(n,k) is the power of 2 in denominator of coefficients of Legendre polynomials, where n >= 0 and 0 <= k <= n.

%I A077070 #41 Jan 29 2022 12:15:24
%S A077070 0,1,1,3,2,3,4,4,4,4,7,5,6,5,7,8,8,7,7,8,8,10,9,10,8,10,9,10,11,11,11,
%T A077070 11,11,11,11,11,15,12,13,12,14,12,13,12,15,16,16,14,14,15,15,14,14,16,
%U A077070 16,18,17,18,15,17,16,17,15,18,17,18,19,19,19,19,18,18,18,18,19,19,19,19,22,20,21,20,22,19,20,19,22,20,21,20,22
%N A077070 Triangle read by rows: T(n,k) is the power of 2 in denominator of coefficients of Legendre polynomials, where n >= 0 and 0 <= k <= n.
%H A077070 Alois P. Heinz, <a href="/A077070/b077070.txt">Rows n = 0..200, flattened</a>
%F A077070 T(n, k) = A007814(A144816(n, k)). - _Michel Marcus_, Jan 29 2022
%F A077070 T(n, k) = 2*n - wt(n-k) - wt(k) where wt = A000120 is the binary weight. - _Kevin Ryde_, Jan 29 2022
%e A077070 Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e A077070    0;
%e A077070    1,  1;
%e A077070    3,  2,  3;
%e A077070    4,  4,  4,  4;
%e A077070    7,  5,  6,  5,  7;
%e A077070    8,  8,  7,  7,  8,  8;
%e A077070   10,  9, 10,  8, 10,  9, 10;
%e A077070   ...
%p A077070 T:= n-> (p-> seq(padic[ordp](denom(coeff(p, x, i)), 2)
%p A077070              , i=0..2*n, 2))(orthopoly[P](2*n, x)):
%p A077070 seq(T(n), n=0..12);  # _Alois P. Heinz_, Jan 25 2022
%t A077070 T[n_, k_] := IntegerExponent[Denominator[Coefficient[LegendreP[2n, x], x, 2k]], 2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2017 *)
%o A077070 (PARI) {T(n, k) = if( k<0 || k>n, 0, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}
%o A077070 (PARI) T(n,k) = 2*n - hammingweight(n-k) - hammingweight(k); \\ _Kevin Ryde_, Jan 29 2022
%Y A077070 Cf. A005187 (column k=0), A101925 (column k=1), A077071 (row sums), A144816 (denominators).
%Y A077070 Cf. A000120, A007814.
%K A077070 nonn,tabl
%O A077070 0,4
%A A077070 _Michael Somos_, Oct 25 2002