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

1, 1, 1, 1, 13, 1, 1, 33, 33, 1, 1, 61, 141, 61, 1, 1, 97, 397, 397, 97, 1, 1, 141, 897, 1597, 897, 141, 1, 1, 193, 1761, 4897, 4897, 1761, 193, 1, 1, 253, 3133, 12541, 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

Offset

0,5

Links

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Formula

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

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

Example

Triangle begins as:
  1;
  1,   1;
  1,  13,    1;
  1,  33,   33,     1;
  1,  61,  141,    61,      1;
  1,  97,  397,   397,     97,      1;
  1, 141,  897,  1597,    897,    141,      1;
  1, 193, 1761,  4897,   4897,   1761,    193,     1;
  1, 253, 3133, 12541,  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;

Mathematica

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

Prog

(Magma)
A141596:= func< n, k | 4*Binomial(n, k)^2 - 3 >;
[A141596(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024
(SageMath)
def A141596(n, k): return 4*binomial(n, k)^2 -3
flatten([[A141596(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024

Crossrefs

Cf. A109128.

Sequence in context: A066834 A010225 A060361 * A108477 A176204 A176492

Adjacent sequences: A141593 A141594 A141595 * A141597 A141598 A141599

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

Status

approved