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

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A134759 (row sums), A141596.

Sequence in context: A081580 A082110 A275526 * A372066 A176561 A176284

Adjacent sequences: A141594 A141595 A141596 * A141598 A141599 A141600

Keyword

nonn,tabl

Author

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

Status

approved