A026637 - OEIS

%I #31 Jun 30 2024 09:10:04

%S 1,1,1,1,2,1,1,4,4,1,1,5,8,5,1,1,7,13,13,7,1,1,8,20,26,20,8,1,1,10,28,

%T 46,46,28,10,1,1,11,38,74,92,74,38,11,1,1,13,49,112,166,166,112,49,13,

%U 1,1,14,62,161,278,332,278,161,62,14,1,1,16,76,223,439,610,610,439,223,76,16,1

%N 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-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.

%C T(n, k) = number of paths from (0, 0) to (n-k, k) in 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 >= 1 and odd and for j=0, i >= 1 and odd.

%C See A228053 for a sequence with many terms in common with this one. - _T. D. Noe_, Aug 07 2013

%H Reinhard Zumkeller, <a href="/A026637/b026637.txt">Rows n = 0..100 of table, flattened</a>

%F From _G. C. Greubel_, Jun 28 2024: (Start)

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

%F T(2*n-1, n-1) = A026641(n), n >= 1.

%F Sum_{k=0..n} T(n, k) = A026644(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  4,  4,   1;

%e   1,  5,  8,   5,   1;

%e   1,  7, 13,  13,   7,   1;

%e   1,  8, 20,  26,  20,   8,   1;

%e   1, 10, 28,  46,  46,  28,  10,   1;

%e   1, 11, 38,  74,  92,  74,  38,  11,  1;

%e   1, 13, 49, 112, 166, 166, 112,  49, 13,  1;

%e   1, 14, 62, 161, 278, 332, 278, 161, 62, 14,  1;

%p A026637 := proc(n,k)

%p       option remember;

%p       if k=0 or k=n then

%p         1

%p     elif k=1 or k=n-1 then

%p         floor((3*n-1)/2) ;

%p     elif k <0 or k > n then

%p         0;

%p     else

%p         procname(n-1,k-1)+procname(n-1,k) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Apr 26 2015

%t T[n_, k_] := T[n, k] = Which[k == 0 || k == n, 1, k == 1 || k == n-1, Floor[(3n-1)/2], k < 0 || k > n, 0, True, T[n-1, k-1] + T[n-1, k]];

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 30 2018 *)

%o (Haskell)

%o a026637 n k = a026637_tabl !! n !! k

%o a026637_row n = a026637_tabl !! n

%o a026637_tabl = [1] : [1,1] : map (fst . snd)

%o    (iterate f (0, ([1,2,1], [0,1,1,0]))) where

%o    f (i, (xs, ws)) = (1 - i,

%o      if i == 1 then (ys, ws) else (zipWith (+) ys ws, ws'))

%o         where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])

%o               ws' = [0,1,0,0] ++ drop 2 ws

%o -- _Reinhard Zumkeller_, Aug 08 2013

%o (Magma)

%o function T(n,k) // T = A026637

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

%o    elif k eq 1 or k eq n-1 then return Floor((3*n-1)/2);

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

%o    end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 28 2024

%o (SageMath)

%o def T(n,k): # T = A026637

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

%o     elif k==1 or k==n-1: return ((3*n-1)//2)

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

%o flatten([[T(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 28 2024

%Y Cf. A026638, A026639, A026640, A026641, A026642, A026643, A026644.

%Y Cf. A026966, A026967, A026968, A026969, A026970, A228053.

%Y Sums include: A000007 (alternating sign row), A026644 (row), A026645, A026646, A026647 (diagonal).

%K nonn,tabl,easy

%O 0,5

%A _Clark Kimberling_