cp's OEIS Frontend

A141596 Triangle T(n,k) = 4*binomial(n,k)^2 - 3, read by rows, 0<=k<=n.

%I A141596 #15 Sep 15 2024 02:11:42
%S A141596 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 A141596 1597,897,141,1,1,193,1761,4897,4897,1761,193,1,1,253,3133,12541,
%U A141596 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 A141596 Triangle T(n,k) = 4*binomial(n,k)^2 - 3, read by rows, 0<=k<=n.
%H A141596 Harvey P. Dale, <a href="/A141596/b141596.txt">Table of n, a(n) for n = 0..10000</a>
%F A141596 Sum_{k=0..n} T(n, k) = 4*binomial(2*n,n) - 3*(n+1) (row sums).
%F A141596 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 A141596 Triangle begins as:
%e A141596   1;
%e A141596   1,   1;
%e A141596   1,  13,    1;
%e A141596   1,  33,   33,     1;
%e A141596   1,  61,  141,    61,      1;
%e A141596   1,  97,  397,   397,     97,      1;
%e A141596   1, 141,  897,  1597,    897,    141,      1;
%e A141596   1, 193, 1761,  4897,   4897,   1761,    193,     1;
%e A141596   1, 253, 3133, 12541,  19597,  12541,   3133,   253,    1;
%e A141596   1, 321, 5181, 28221,  63501,  63501,  28221,  5181,  321,   1;
%e A141596   1, 397, 8097, 57597, 176397, 254013, 176397, 57597, 8097, 397,  1;
%t A141596 Table[4*Binomial[n,k]^2-3,{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Dec 21 2016 *)
%o A141596 (Magma)
%o A141596 A141596:= func< n,k | 4*Binomial(n,k)^2 - 3 >;
%o A141596 [A141596(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 15 2024
%o A141596 (SageMath)
%o A141596 def A141596(n,k): return 4*binomial(n,k)^2 -3
%o A141596 flatten([[A141596(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 15 2024
%Y A141596 Cf. A109128.
%K A141596 nonn,tabl
%O A141596 0,5
%A A141596 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 21 2008