A028314 Elements in the 5-Pascal triangle A028313 that are not 1.

5, 6, 6, 7, 12, 7, 8, 19, 19, 8, 9, 27, 38, 27, 9, 10, 36, 65, 65, 36, 10, 11, 46, 101, 130, 101, 46, 11, 12, 57, 147, 231, 231, 147, 57, 12, 13, 69, 204, 378, 462, 378, 204, 69, 13, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 15, 96, 355, 855, 1422, 1680, 1422, 855, 355, 96, 15

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
   5;
   6,  6;
   7, 12,   7;
   8, 19,  19,   8;
   9, 27,  38,  27,   9;
  10, 36,  65,  65,  36,  10;
  11, 46, 101, 130, 101,  46,  11;
  12, 57, 147, 231, 231, 147,  57,  12;
  13, 69, 204, 378, 462, 378, 204,  69,  13;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000027, A010892, A022112, A028313, A028322,

Cf. A051936, A051937, A121262, A176448.

Programs

Magma

A028314:= func< n,k | Binomial(n+2,k+1) + 3*Binomial(n,k) >;
[A028314(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 06 2024

Mathematica

A028314[n_, k_]:= Binomial[n+2,k+1] + 3*Binomial[n,k];
Table[A028314[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 06 2024 *)

SageMath

def A028314(n,k): return binomial(n+2,k+1) + 3*binomial(n,k)
flatten([[A028314(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 06 2024

Formula

From G. C. Greubel, Jan 06 2024: (Start)

T(n, k) = binomial(n+2, k+1) + 3*binomial(n, k).

T(n, n-k) = T(n, k).

T(n, 0) = T(n, n) = A000027(n+5).

T(n, 1) = T(n, n-1) = A051936(n+4).

T(n, 2) = T(n, n-2) = A051937(n+3).T(2*n, n) = A028322(n+1).

Sum_{k=0..n} T(n, k) = A176448(n).

Sum_{k=0..n} (-1)^k * T(n, k) = 1 + (-1)^n + 3*[n=0].

Sum_{k=0..n} T(n-k, k) = A022112(n+1) - (3-(-1)^n)/2.

Sum_{k=0..n} (-1)^k * T(n-k, k) = 4*A010892(n) - 2*A121262(n+1) - (3 - (-1)^n)/2. (End)

G.f.: (5 - 4*x - 4*x*y + 3*x^2*y)/((1 - x)*(1 - x*y)*(1 - x - x*y)). - Stefano Spezia, Dec 06 2024

Extensions

More terms from James Sellers