A141597 Triangle T(n,k) = 2*binomial(n,k)^2 - 1, read by rows, 0<=k<=n.
1, 1, 1, 1, 7, 1, 1, 17, 17, 1, 1, 31, 71, 31, 1, 1, 49, 199, 199, 49, 1, 1, 71, 449, 799, 449, 71, 1, 1, 97, 881, 2449, 2449, 881, 97, 1, 1, 127, 1567, 6271, 9799, 6271, 1567, 127, 1, 1, 161, 2591, 14111, 31751, 31751, 14111, 2591, 161, 1, 1, 199, 4049, 28799, 88199, 127007, 88199, 28799, 4049, 199, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 7, 1; 1, 17, 17, 1; 1, 31, 71, 31, 1; 1, 49, 199, 199, 49, 1; 1, 71, 449, 799, 449, 71, 1; 1, 97, 881, 2449, 2449, 881, 97, 1; 1, 127, 1567, 6271, 9799, 6271, 1567, 127, 1; 1, 161, 2591, 14111, 31751, 31751, 14111, 2591, 161, 1; 1, 199, 4049, 28799, 88199, 127007, 88199, 28799, 4049, 199, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
A141597:= func< n,k | 2*Binomial(n,k)^2 - 1 >; [A141597(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024
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Mathematica
T[n_, m_, k_, l_]:= (1+l)*Binomial[n, m]^k -l; Table[T[n,m,2,1], {n,0,12}, {m,0,n}]//Flatten -
SageMath
def A141597(n,k): return 2*binomial(n,k)^2 -1 flatten([[A141597(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024
Formula
Sum_{k=0..n} T(n, k) = A134759(n) = 2*binomial(2*n,n) - (n+1) (row sums).
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = ((1+(-1)^n)/2)*(2*(-1)^(n/2)*binomial(n, n/2) - 1) (alternating sign row sums). - G. C. Greubel, Sep 15 2024