A326832 Triangle read by rows: T(n, k) = (-1)^(n - k) * binomial(2n + 1, n - k) * L(2k + 1), 0 <= k <= n, where L(k) is the k-th Lucas number (A000032).

1, -3, 4, 10, -20, 11, -35, 84, -77, 29, 126, -336, 396, -261, 76, -462, 1320, -1815, 1595, -836, 199, 1716, -5148, 7865, -8294, 5928, -2587, 521, -6435, 20020, -33033, 39585, -34580, 20895, -7815, 1364, 24310, -77792, 136136, -179452, 180880, -135320, 70856, -23188, 3571

Offset: 0

Author

Amiram Eldar, Oct 20 2019

Keywords

• sign
• tabl

Comments

This triangle is the subject of the paper by Agarwal (1990).

Links

• Ashok Agarwal, On a new kind of numbers, Fibonacci Quarterly, Vol. 28, No. 3 (1990), pp. 194-199.
• R. K. Raina and H. M. Srivastava, A class of numbers associated with the Lucas numbers, Mathematical and Computer Modelling, Vol. 25, No. 7 (1997), pp. 15-22.

Crossrefs

Cf. A000032, A007318.

Programs

Mathematica

T[n_, k_] := (-1)^(n - k) * Binomial[2 n + 1, n - k] * LucasL[2 k + 1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

PARI

L(n) = fibonacci(n+1) + fibonacci(n-1); \\ A000032
T(n, k) = (-1)^(n - k) * binomial(2*n + 1, n - k) * L(2*k + 1); \\ Michel Marcus, Oct 21 2019

Formula

Sum_{k=0..n} T(n, k) = 1.

T(n + 1, k) = -(2n + 3)*(2n + 2)/((n - k + 1)*(n + k + 2)) * T(n, k).