A028313 - OEIS

%I #21 Jan 07 2024 02:16:21

%S 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,

%T 36,65,65,36,10,1,1,11,46,101,130,101,46,11,1,1,12,57,147,231,231,147,

%U 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

%N Elements in the 5-Pascal triangle (by row).

%H G. C. Greubel, <a href="/A028313/b028313.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _Ralf Stephan_, Jan 31 2005: (Start)

%F T(n, k) = C(n, k) + 3*C(n-2, k-1), with T(0, k) = T(1, k) = 1.

%F G.f.: (1 + 3*x^2*y)/(1 - x*(1+y)). (End)

%F From _G. C. Greubel_, Jan 05 2024: (Start)

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

%F T(n, n-1) = n + 3*(1 - [n=1]) = A178915(n+3), n >= 1.

%F T(n, n-2) = A051936(n+2), n >= 2.

%F T(n, n-3) = A051937(n+1), n >= 3.

%F T(2*n, n) = A028322(n).

%F Sum_{k=0..n} T(n, k) = A005009(n-2) - (3/4)*[n=0] - (3/2)*[n=1].

%F Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n) - 3*[n=2].

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A022112(n-2) + 3*([n=0] - [n=1]).

%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = 4*A010892(n) - 3*([n=0] + [n=1]). (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  5,  1;

%e   1,  6,  6,   1;

%e   1,  7, 12,   7,   1;

%e   1,  8, 19,  19,   8,   1;

%e   1,  9, 27,  38,  27,   9,   1;

%e   1, 10, 36,  65,  65,  36,  10,  1;

%e   1, 11, 46, 101, 130, 101,  46, 11,  1;

%e   1, 12, 57, 147, 231, 231, 147, 57, 12,  1;

%t 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 *)

%o (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

%o (SageMath)

%o def A028313(n,k): return 1 if n<2 else binomial(n,k) + 3*binomial(n-2,k-1)

%o flatten([[A028313(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 05 2024

%Y Cf. A000007, A005009, A010892, A022112, A028275, A028314, A028315.

%Y Cf. A028316, A028317, A028318, A028319, A028320, A028321, A028322.

%Y Cf. A028323, A028324, A028325, A029653, A051472, A051936, A051937.

%Y Cf. A178915.

%K nonn,tabl

%O 0,5

%A _Mohammad K. Azarian_

%E More terms from Sam Alexander (pink2001x(AT)hotmail.com)