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

Author

Christian Barrientos and Sarah Minion, Jul 29 2018

Keywords

• nonn
• tabf

Examples

			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;
		

Crossrefs

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

Programs

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 *)

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

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.