Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Nov 20 2021 07:19:49
%S 1,1,6,57,629,7589,96942,1288729,17643920,247089010,3522891561,
%T 50964747400,746241617226,11038241689188,164696773030055,
%U 2475832560808858,37462189433509758,570112127356828846,8720472842436039280,133997057207982607092,2067402314984991892461
%N G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).
%H Seiichi Manyama, <a href="/A349363/b349363.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
%F a(n) = (-1)^(n+1)* F([8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 1-n], [4/3, 3/2, 5/3, 11/6, 2, 13/6], 7^7/6^6), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 15 2021
%F a(n) ~ 776887^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - _Vaclav Kotesovec_, Nov 17 2021
%p a:= n-> coeff(series(RootOf(1+x*A^7/(1+x)-A, A), x, n+1), x, n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Nov 15 2021
%t nmax = 20; A[_] = 0; Do[A[x_] = 1 + x A[x]^7/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
%Y Cf. A001006, A002296, A127897, A317133, A346667, A346671 (binomial transform), A349334, A349361, A349362, A349364.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 15 2021