A077071 -id:A077071 - cp's OEIS frontend

A076178 a(n) = 2*n^2 - A077071(n).

0, 0, 0, 2, 2, 4, 6, 10, 10, 12, 14, 18, 20, 24, 28, 34, 34, 36, 38, 42, 44, 48, 52, 58, 60, 64, 68, 74, 78, 84, 90, 98, 98, 100, 102, 106, 108, 112, 116, 122, 124, 128, 132, 138, 142, 148, 154, 162, 164, 168, 172, 178, 182, 188, 194, 202, 206, 212, 218, 226, 232

Offset: 0

Benoit Cloitre, Nov 01 2002

nonn

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).

Equals 2 * A078903(n).

Cf. A001105, A067699, A077070, A077071.

a(n)=2*n^2-sum(k=0,n,-valuation(polcoeff(pollegendre(2*n),2*k),2))

def A076178(n): return ((n+1)*n.bit_count()-n<<1)+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 12 2024

a(n) = A001105(n) - A077071(n). - Omar E. Pol, Nov 13 2024

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.

Original entry on oeis.org

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, 11, 11, 11, 11, 11, 15, 12, 13, 12, 14, 12, 13, 12, 15, 16, 16, 14, 14, 15, 15, 14, 14, 16, 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

Offset: 0

Michael Somos, Oct 25 2002

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
   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;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A005187 (column k=0), A101925 (column k=1), A077071 (row sums), A144816 (denominators).

Cf. A000120, A007814.

T:= n-> (p-> seq(padic[ordp](denom(coeff(p, x, i)), 2)
             , i=0..2*n, 2))(orthopoly[P](2*n, x)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jan 25 2022

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 *)

{T(n, k) = if( k<0 || k>n, 0, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}

T(n,k) = 2*n - hammingweight(n-k) - hammingweight(k); \\ Kevin Ryde, Jan 29 2022

T(n, k) = A007814(A144816(n, k)). - Michel Marcus, Jan 29 2022

T(n, k) = 2*n - wt(n-k) - wt(k) where wt = A000120 is the binary weight. - Kevin Ryde, Jan 29 2022

cp's OEIS Frontend

A076178 a(n) = 2*n^2 - A077071(n).

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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.

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