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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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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
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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 *) -
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
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)
Extensions
More terms from Sam Alexander (pink2001x(AT)hotmail.com)