A324242 Incomplete Lucas numbers: irregular triangular array L(n,k) = Sum_{j = 0..k} (n/(n-j)) * binomial(n-j, j), read by rows, with n >= 1 and 0 <= k <= floor(n/2).

1, 1, 3, 1, 4, 1, 5, 7, 1, 6, 11, 1, 7, 16, 18, 1, 8, 22, 29, 1, 9, 29, 45, 47, 1, 10, 37, 67, 76, 1, 11, 46, 96, 121, 123, 1, 12, 56, 133, 188, 199, 1, 13, 67, 179, 284, 320, 322, 1, 14, 79, 235, 417, 508, 521, 1, 15, 92, 302, 596, 792, 841, 843, 1, 16, 106, 381, 831, 1209, 1349, 1364

Offset

1,3

Comments

For additional properties of the incomplete Lucas numbers and special cases not listed here, see Filipponi (1996, pp. 45-53).

Links

Table of n, a(n) for n=1..71.

A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Lucas numbers.

Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 2 (p. 46) that contains the incomplete Lucas numbers.

A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.

Formula

L(n,k) = F(n-1, k-1) + F(n+1, k) for n >= 1 and 0 <= k <= floor(n/2), where F(n,k) = Sum_{j = 0..k} binomial(n-1-j, j) are the incomplete Fibonacci numbers (defined for n >= 1 and 0 <= k <= floor((n-1)/2)).

L(n+2, k+1) = L(n+1, k+1) + L(n,k) for n >= 1 and 0 <= k <= floor((n-1)/2).

L(n,k) = F(n+2,k) - F(n-2, k-2) for n >= 3 and 2 <= k <= floor((n+1)/2).

Special cases: L(n,0) = 1 (n >= 1), L(n,1) = n+1 (n >= 2), L(n,2) = (n^2-n+2)/2 = A000124(n-1) (n >= 4), and L(n, floor(n/2)) = A000204(n) (n >= 1).

Sum of row n = (3 + (-1)^n)*A000204(n)/4 + n*A000045(n)/2.

G.f. for column k >= 1: t^(2*k)*((A000204(2*k) + t*A000204(2*k-1))*(1-t)^(k+1) - t^2*(2-t))/((1-t)^(k+1) * (1-t-t^2)).

Example

Triangle L(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  4;
  1,  5,  7;
  1,  6, 11;
  1,  7, 16,  18;
  1,  8, 22,  29;
  1,  9, 29,  45,  47;
  1, 10, 37,  67,  76;
  1, 11, 46,  96, 121, 123;
  1, 12, 56, 133, 188, 199;
  ...
Row sums are 1, 4, 5, 13, 18, 42, 60, 131, 191, 398, 589, 1186, 1775, 3482, 5257, 10103, 15360, ...

Mathematica

Flatten[Table[Sum[(n/(n-j))*Binomial[n-j, j], {j, 0, k}], {n, 1, 15}, {k, 0, Floor[n/2]}]] (* Stefano Spezia, Sep 03 2019 *)

Crossrefs

Cf. A038730, A038792, and A134511 for various versions of the incomplete Fibonacci numbers.

Cf. A000032, A000045, A000124, A000204, A137176.

Sequence in context: A327642 A358919 A354617 * A216543 A152040 A013705

Adjacent sequences: A324239 A324240 A324241 * A324243 A324244 A324245

Keyword

nonn,tabf

Author

Petros Hadjicostas, Sep 02 2019

Status

approved