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URL: https://oeis.org/A026626

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A026626
Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor(3*n/2) for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.
16
1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 21, 28, 21, 9, 1, 1, 10, 30, 49, 49, 30, 10, 1, 1, 12, 40, 79, 98, 79, 40, 12, 1, 1, 13, 52, 119, 177, 177, 119, 52, 13, 1, 1, 15, 65, 171, 296, 354, 296, 171, 65, 15, 1, 1, 16, 80, 236, 467, 650, 650, 467, 236, 80, 16, 1
OFFSET
0,5
FORMULA
T(n, k) = number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=0, j >= 0 and even and for j=0, i >= 0 and even.
From G. C. Greubel, Jun 19 2024: (Start)
T(n, n-k) = T(n, k)
T(n, 1) = (-1)^n*A084056(n) = A032766(n), n >= 1.
T(n, 2) = A006578(n-1), n >= 2.
T(n, 3) = (1/16)*(4*n^3 - 14*n^2 + 12*n + 15 + (-1)^n) - [n=3] , n >= 3.
Sum_{k=0..n} (-1)^k*T(n, k) = A176742(n) + [n=2].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/4)*((-1)^n*(8/sqrt(3)* sin(2*(n+1)*Pi/3) - 2*cos(n*Pi/2) + 1) - 3) + [n<2].
Sum_{k=0..n} k*T(n, k) = (1/6)*n*(17*2^(n-2) - 2 - (1-(-1)^n)) + (1/4)*[n=1]. (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 4, 4, 1;
1, 6, 8, 6, 1;
1, 7, 14, 14, 7, 1;
1, 9, 21, 28, 21, 9, 1;
1, 10, 30, 49, 49, 30, 10, 1;
1, 12, 40, 79, 98, 79, 40, 12, 1;
1, 13, 52, 119, 177, 177, 119, 52, 13, 1;
1, 15, 65, 171, 296, 354, 296, 171, 65, 15, 1;
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, (6*n-1+(-1)^n)/4, T[n-1, k-1] + T[n-1, k] ]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 19 2024 *)
PROG
(Magma)
function T(n, k) // T = A026626
if k eq 0 or k eq n then return 1;
elif k eq 1 or k eq n-1 then return Floor(3*n/2);
else return T(n-1, k-1) + T(n-1, k);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2024
(SageMath)
def T(n, k): # T = A026626
if (k==0 or k==n): return 1
elif (k==1 or k==n-1): return int(3*n//2)
else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 19 2024
KEYWORD
nonn,tabl,easy
STATUS
approved