A216703 a(n) = Product_{k=1..n} (49 - 7/k).

1, 42, 1911, 89180, 4213755, 200574738, 9594158301, 460519598448, 22162505675310, 1068725273676060, 51619430718553698, 2496503376570051576, 120872371815599997138, 5857661095679076784380, 284096563140435224042430, 13788153197749122873525936

Offset: 0

Author

Michel Lagneau, Sep 16 2012

Keywords

• nonn
• easy

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A004994, A049382, A216702, A220607.

Cf. A034835, A386271, A386272, A386273, A386274.

Programs

Maple

seq(product(49-7/k, k=1.. n), n=0..20);
seq((7^n/n!)*product(7*k+6, k=0.. n-1), n=0..20);

Mathematica

Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

Formula

From Seiichi Manyama, Jul 17 2025: (Start)

G.f.: 1/(1 - 49*x)^(6/7).

a(n) = (-49)^n * binomial(-6/7,n).

a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+6). (End)

From Amiram Eldar, Aug 17 2025: (Start)

a(n) = 49^n * Gamma(n+6/7) / (Gamma(6/7) * Gamma(n+1)).

a(n) ~ c * 49^n / n^(1/7), where c = 1/Gamma(6/7) = 1/A220607 = 0.904349... . (End)