A034835 Expansion of 1/(1-49*x)^(1/7); related to sept-factorial numbers A045754.

1, 7, 196, 6860, 264110, 10722866, 450360372, 19365495996, 847240449825, 37560993275575, 1682732498745760, 76028913806967520, 3459315578217022160, 158330213003009860400, 7283189798138453578400, 336483368673996555322080, 15604416222256590253061460, 726064307753233111186565580

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009), 31-41, eq (1.10).
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A045754, A004993, A034829, A034830, A034831, A034832, A034833, A034834, A220086.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1 - 49*x)^(1/7))); // G. C. Greubel, Feb 22 2018

Mathematica

CoefficientList[Series[1/(1 - 49*x)^(1/7), {x,0,50}], x] (* G. C. Greubel, Feb 22 2018 *)

PARI

my(x='x+O('x^30)); Vec(1/(1 - 49*x)^(1/7)) \\ G. C. Greubel, Feb 22 2018

Formula

a(n) = 7^n*A045754(n)/n!, n >= 1, where A045754(n) = (7*n-6)(!^7) = Product_{j=1..n} (7*j-6).

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

D-finite with recurrence: n*a(n) + 7*(-7*n+6)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020

a(n) ~ 7^(2*n) * n^(-6/7) / Gamma(1/7). - Amiram Eldar, Aug 18 2025