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A028314
Elements in the 5-Pascal triangle A028313 that are not 1.
7
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
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
KEYWORD
nonn,tabl
EXTENSIONS
More terms from James A. Sellers
STATUS
approved