A317489 Irregular triangle read by rows. For n >= 3 and 1 <= k <= floor(n/3), T(n,k) is the number of palindromic compositions of n into k parts of size at least 3.

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 3, 2, 1, 0, 4, 0, 1, 1, 1, 4, 3, 1, 1, 0, 5, 0, 3, 1, 1, 5, 4, 3, 1, 1, 0, 6, 0, 6, 0, 1, 1, 6, 5, 6, 3, 1, 0, 7, 0, 10, 0, 1, 1, 1, 7, 6, 10, 6, 1, 1, 0, 8, 0, 15, 0, 4, 1, 1, 8, 7, 15, 10, 4, 1, 1, 0, 9, 0, 21, 0, 10, 0, 1, 1, 9, 8, 21, 15, 10, 4, 1, 0, 10, 0, 28, 0, 20, 0, 1, 1, 1, 10, 9, 28, 21, 20, 10, 1, 1, 0, 11, 0, 36, 0, 35, 0, 5

Offset

3,18

Links

Table of n, a(n) for n=3..137.

Formula

T(n,k) = 0 if n is odd and k is even;

T(n,k) = binomial((n-1)/2-k,(k-1)/2) if n is odd and k is odd;

T(n,k) = binomial((n-2)/2-k,(k-1)/2) if n is even and k is odd;

T(n,k) = binomial((n-2)/2-k,(k-2)/2) if n is even and k is even.

Example

For n=24 and k=3, T(24,3) = 8 = binomial((24-2)/2-3, (3-1)/2) = binomial(8,1).
The first entries of the irregular triangle formed by the values of T(n,k) are:
  1;
  1;
  1;
  1,  1;
  1,  0;
  1,  1;
  1,  0,  1;
  1,  1,  1;
  1,  0,  2;
  1,  1,  2,  1;
  1,  0,  3,  0;
  1,  1,  3,  2;
  1,  0,  4,  0,  1;
  1,  1,  4,  3,  1;
  1,  0,  5,  0,  3;
  1,  1,  5,  4,  3,  1;
  1,  0,  6,  0,  6,  0;
  1,  1,  6,  5,  6,  3;
  1,  0,  7,  0, 10,  0,  1;
  1,  1,  7,  6, 10,  6,  1;
  1,  0,  8,  0, 15,  0,  4;
  1,  1,  8,  7, 15, 10,  4,  1;
  1,  0,  9,  0, 21,  0, 10,  0;
  1,  1,  9,  8, 21, 15, 10,  4;
  1,  0, 10,  0, 28,  0, 20,  0,  1;
  1,  1, 10,  9, 28, 21, 20, 10,  1;
  1,  0, 11,  0, 36,  0, 35,  0,  5;

Mathematica

T[n_, k_] := If[Mod[n, 2] == 1 && Mod[k, 2] == 0, 0, Binomial[Quotient[n-1, 2] - k, Quotient[k-1, 2]]];
Table[T[n, k], {n, 3, 30}, {k, 1, Quotient[n, 3]}] // Flatten (* Jean-François Alcover, Sep 13 2018, from PARI *)

Prog

(PARI) T(n, k)=if(n%2==1&&k%2==0, 0,  binomial((n-1)\2-k, (k-1)\2)); \\ Andrew Howroyd, Sep 07 2018

Crossrefs

Row sums of the triangle equal A226916(n+4).

Sequence in context: A143158 A336708 A308424 * A345647 A091950 A014750

Adjacent sequences: A317486 A317487 A317488 * A317490 A317491 A317492

Keyword

nonn,tabf

Author

Christian Barrientos and Sarah Minion, Jul 29 2018

Status

approved