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

0,1

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A051936, A051937, A121262, A176448.

Sequence in context: A335015 A195875 A205852 * A201326 A232247 A200110

Adjacent sequences: A028311 A028312 A028313 * A028315 A028316 A028317

Keyword

nonn,tabl

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved