A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

1, 1, 4, 40, 1120, 91840, 22408960, 16358540800, 35792487270400, 234870301468364800, 4623187014103292723200, 272999193182799435304960000, 48361261073946554365403054080000, 25701205307660304745058529866383360000, 40976048450930207702360695570691784048640000

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..50

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(3*x) / (1 - x).

G.f.: Sum_{k>=0} 3^(k*(k - 1)/2) * x^k / Product_{j=0..k-1} (1 - 3^j*x).

a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * a(k).

a(n) ~ c * 3^(n*(n - 1)/2), where c = Product_{k>=1} (1 + 1/3^k) = 1.564934018567011537938849... = A132324.

a(n) = 3^(binomial(n+1,2))*(-1/3;1/3)_{n}, where (a;q)_{n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

Mathematica

Table[Product[3^k + 1, {k, 1, n - 1}], {n, 0, 14}]

Prog

(PARI) a(n) = prod(k=1, n-1, 3^k + 1); \\ Michel Marcus, Jun 06 2020
(Sage)
from sage.combinat.q_analogues import q_pochhammer
[1]+[3^(binomial(n, 2))*q_pochhammer(n-1, -1/3, 1/3) for n in (1..20)] # G. C. Greubel, Feb 21 2021
(Magma)
[n lt 3 select 1 else (&*[3^j +1: j in [1..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 21 2021

Crossrefs

Cf. A027871, A034472, A047656, A132324, A323716.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), this sequence (m=3), A309327 (m=4).

Sequence in context: A013108 A173945 A111846 * A363423 A102922 A139688

Adjacent sequences: A289997 A289998 A289999 * A290001 A290002 A290003

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 06 2020

Status

approved