A131404 - OEIS

%I #18 Sep 22 2024 02:02:16

%S 1,1,1,1,5,1,1,7,7,1,1,11,13,11,1,1,13,27,27,13,1,1,17,39,65,39,17,1,

%T 1,19,61,111,111,61,19,1,1,23,79,193,221,193,79,23,1,1,25,109,283,433,

%U 433,283,109,25,1,1,29,133,425,715,925,715,425,133,29,1

%N a(n) = 2*A131402(n) - 1.

%C Row sums = A131405: (1, 2, 7, 16, 37, 82, 179, ...).

%H Andrew Howroyd, <a href="/A131404/b131404.txt">Table of n, a(n) for n = 0..1274</a>

%F T(n,k) = 4*binomial(n, k) + 1 - 2*binomial(floor((n + k)/2), k) - 2*binomial(n-floor((k+1)/2), floor(k/2)). - _Andrew Howroyd_, Aug 09 2018

%e First few rows of the triangle are:

%e   1;

%e   1,  1;

%e   1,  5,  1;

%e   1,  7,  7,  1;

%e   1, 11, 13, 11,  1;

%e   1, 13, 27, 27, 13,  1;

%e   1, 17, 39, 65, 39, 17,  1;

%e   ...

%o (PARI) T(n,k) = if(k <= n, 4*binomial(n,k) + 1 - 2*binomial((n + k)\2, k) - 2*binomial(n-(k+1)\2, k\2), 0) \\ _Andrew Howroyd_, Aug 09 2018

%o (Magma) /* As triangle */ [[4*Binomial(n, k) + 1 - 2*Binomial(Floor(n + k) div 2, k) - 2*Binomial(n-Floor((k+1)/2), Floor(k/2)): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 10 2018

%Y Row sums are A131405.

%Y Cf. A131402.

%K nonn,tabl,easy

%O 0,5

%A _Gary W. Adamson_, Jul 07 2007

%E Terms a(55) and beyond from _Andrew Howroyd_, Aug 09 2018