A386271 Expansion of 1/(1 - 49*x)^(2/7).

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145

Offset: 0

Author

Seiichi Manyama, Jul 17 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

Formula

a(n) = (-49)^n * binomial(-2/7,n).

a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).

a(n) = 7^n * Product_{k=1..n} (7 - 5/k).

In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025