A026615 - OEIS

%I #19 Jun 15 2024 02:14:34

%S 1,1,1,1,3,1,1,5,5,1,1,7,10,7,1,1,9,17,17,9,1,1,11,26,34,26,11,1,1,13,

%T 37,60,60,37,13,1,1,15,50,97,120,97,50,15,1,1,17,65,147,217,217,147,

%U 65,17,1,1,19,82,212,364,434,364,212,82,19,1

%N Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0,  T(n,1) = T(n,n-1) = 2*n-1 for n >= 1, T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2 and n >= 4.

%C T(n, k) is the 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 for j=0, i >= 0.

%H G. C. Greubel, <a href="/A026615/b026615.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A026622(n) (row sums).

%F From _G. C. Greubel_, Jun 13 2024: (Start)

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

%F T(n, 0) = A000012(n).

%F T(n, 1) = A005408(n-1), n >= 1.

%F T(n, 2) = A098749(n), n >= 2.

%F T(n, 3) = A145066(n-2) - [n=3], n >= 3.

%F Sum_{k=0..n} (-1)^k*T(n, k) = A176742(n) + [n=2].

%F Sum_{k=0..n} (-1)^k*T(n-k, k) = b(n-2) + 2*[n=0] + [n=1], where b(n) = (1/6)*(-2*sqrt(3)*sin(Pi*n/3) + 2*sqrt(3)*sin(5*Pi*n/3) + 3*cos(Pi* n/2) + 3*cos(3*Pi*n/2) - 6).

%F Sum_{k=0..n} k*T(n, k) = n*(7*2^(n-3) - 1) + (1/4)*[n=1]. (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  3,  1;

%e   1,  5,  5,   1;

%e   1,  7, 10,   7,   1;

%e   1,  9, 17,  17,   9,   1;

%e   1, 11, 26,  34,  26,  11,   1;

%e   1, 13, 37,  60,  60,  37,  13,   1;

%e   1, 15, 50,  97, 120,  97,  50,  15,  1;

%e   1, 17, 65, 147, 217, 217, 147,  65, 17,  1;

%e   1, 19, 82, 212, 364, 434, 364, 212, 82, 19,  1;

%t T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, 2*n-1, T[n-1, k -1] + T[n-1,k]]];

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 13 2024 *)

%o (Magma)

%o function T(n,k) // T = A026615

%o    if k eq 0 or k eq n then return 1;

%o    elif k eq 1 or k eq n-1 then return 2*n-1;

%o    else return T(n-1, k-1) + T(n-1, k);

%o    end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 13 2024

%o (SageMath)

%o def T(n,k): # T = A026615

%o     if k==0 or k==n: return 1

%o     elif k==1 or k==n-1: return 2*n-1

%o     else: return T(n-1, k-1) + T(n-1, k)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jun 13 2024

%Y Cf. A026616, A026617, A026618, A026619, A026620, A026621, A026623.

%Y Cf. A026624, A026956, A026957, A026958, A026959, A026960.

%Y Cf. A000012, A005408, A098749, A145066, A176742.

%Y Cf. A026622 (row sums), A026625 (diagonal sums).

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_

%E Offset corrected by _G. C. Greubel_, Jun 13 2024