A141596 - OEIS

%I #15 Sep 15 2024 02:11:42

%S 1,1,1,1,13,1,1,33,33,1,1,61,141,61,1,1,97,397,397,97,1,1,141,897,

%T 1597,897,141,1,1,193,1761,4897,4897,1761,193,1,1,253,3133,12541,

%U 19597,12541,3133,253,1,1,321,5181,28221,63501,63501,28221,5181,321,1,1,397,8097,57597,176397,254013,176397,57597,8097,397,1

%N Triangle T(n,k) = 4*binomial(n,k)^2 - 3, read by rows, 0<=k<=n.

%H Harvey P. Dale, <a href="/A141596/b141596.txt">Table of n, a(n) for n = 0..10000</a>

%F Sum_{k=0..n} T(n, k) = 4*binomial(2*n,n) - 3*(n+1) (row sums).

%F Sum_{k=0..n} (-1)^k*T(n, k) = ((1 + (-1)^n)/2)*(4*(-1)^(n/2)*binomial(n, n/2) - 3) (alternating sign row sums). - _G. C. Greubel_, Sep 15 2024

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,  13,    1;

%e   1,  33,   33,     1;

%e   1,  61,  141,    61,      1;

%e   1,  97,  397,   397,     97,      1;

%e   1, 141,  897,  1597,    897,    141,      1;

%e   1, 193, 1761,  4897,   4897,   1761,    193,     1;

%e   1, 253, 3133, 12541,  19597,  12541,   3133,   253,    1;

%e   1, 321, 5181, 28221,  63501,  63501,  28221,  5181,  321,   1;

%e   1, 397, 8097, 57597, 176397, 254013, 176397, 57597, 8097, 397,  1;

%t Table[4*Binomial[n,k]^2-3,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Dec 21 2016 *)

%o (Magma)

%o A141596:= func< n,k | 4*Binomial(n,k)^2 - 3 >;

%o [A141596(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 15 2024

%o (SageMath)

%o def A141596(n,k): return 4*binomial(n,k)^2 -3

%o flatten([[A141596(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 15 2024

%Y Cf. A109128.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 21 2008