A294349 Product of first n terms of the binomial transform of the Lucas numbers (A000032).

2, 6, 42, 756, 35532, 4370436, 1407280392, 1186337370456, 2618246576596392, 15128228719573952976, 228844715840995186667952, 9062937281450932377610903056, 939663463215395570775453650652192, 255065069445576619918001465293982953056

Offset

0,1

Links

Table of n, a(n) for n=0..13.

Hideyuki Ohtsuka, Problem H-804, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 55, No. 1 (2017), p. 90; Sums of reciprocals of products of Lucas numbers, Solution to Problem H-804 by the proposer, ibid., Vol. 57, No. 1 (2019), pp. 94-95.

Formula

a(n) ~ c * phi^(n*(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 2.349094356918735309421297337651771419003525539652230102934874983942...

Sum_{n>=0} 1/(phi^(n*(n-3)) * a(n)) = 1, where phi is the golden ratio (A001622) (Ohtsuka, 2017). - Amiram Eldar, Jan 02 2026

Mathematica

Table[Product[Sum[Binomial[m, k]*LucasL[k], {k, 0, m}], {m, 0, n}], {n, 0, 15}]
Table[Product[LucasL[2*k], {k, 0, n}], {n, 0, 15}]

Crossrefs

Cf. A000032, A001622, A005248, A047656, A086619, A135407, A194157.

Sequence in context: A061062 A375528 A270141 * A325782 A374891 A326343

Adjacent sequences: A294346 A294347 A294348 * A294350 A294351 A294352

Keyword

nonn

Author

Vaclav Kotesovec, Oct 29 2017

Status

approved