A028313 Elements in the 5-Pascal triangle (by row).

1, 1, 1, 1, 5, 1, 1, 6, 6, 1, 1, 7, 12, 7, 1, 1, 8, 19, 19, 8, 1, 1, 9, 27, 38, 27, 9, 1, 1, 10, 36, 65, 65, 36, 10, 1, 1, 11, 46, 101, 130, 101, 46, 11, 1, 1, 12, 57, 147, 231, 231, 147, 57, 12, 1, 1, 13, 69, 204, 378, 462, 378, 204, 69, 13, 1, 1, 14, 82, 273, 582, 840, 840, 582, 273, 82, 14, 1

Offset

0,5

Links

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

Formula

From Ralf Stephan, Jan 31 2005: (Start)

T(n, k) = C(n, k) + 3*C(n-2, k-1), with T(0, k) = T(1, k) = 1.

G.f.: (1 + 3*x^2*y)/(1 - x*(1+y)). (End)

From G. C. Greubel, Jan 05 2024: (Start)

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

T(n, n-1) = n + 3*(1 - [n=1]) = A178915(n+3), n >= 1.

T(n, n-2) = A051936(n+2), n >= 2.

T(n, n-3) = A051937(n+1), n >= 3.

T(2*n, n) = A028322(n).

Sum_{k=0..n} T(n, k) = A005009(n-2) - (3/4)*[n=0] - (3/2)*[n=1].

Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n) - 3*[n=2].

Sum_{k=0..floor(n/2)} T(n-k, k) = A022112(n-2) + 3*([n=0] - [n=1]).

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  5,  1;
  1,  6,  6,   1;
  1,  7, 12,   7,   1;
  1,  8, 19,  19,   8,   1;
  1,  9, 27,  38,  27,   9,   1;
  1, 10, 36,  65,  65,  36,  10,  1;
  1, 11, 46, 101, 130, 101,  46, 11,  1;
  1, 12, 57, 147, 231, 231, 147, 57, 12,  1;

Mathematica

Table[If[n<2, 1, Binomial[n, k] +3*Binomial[n-2, k-1]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 05 2024 *)

Prog

(Magma) [n le 1 select 1 else Binomial(n, k) +3*Binomial(n-2, k-1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 05 2024
(SageMath)
def A028313(n, k): return 1 if n<2 else binomial(n, k) + 3*binomial(n-2, k-1)
flatten([[A028313(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2024

Crossrefs

Cf. A000007, A005009, A010892, A022112, A028275, A028314, A028315.

Cf. A028316, A028317, A028318, A028319, A028320, A028321, A028322.

Cf. A028323, A028324, A028325, A029653, A051472, A051936, A051937.

Cf. A178915.

Sequence in context: A159570 A280374 A259975 * A173119 A050178 A297986

Adjacent sequences: A028310 A028311 A028312 * A028314 A028315 A028316

Keyword

nonn,tabl

Author

Mohammad K. Azarian

Extensions

More terms from Sam Alexander (pink2001x(AT)hotmail.com)

Status

approved