cp's OEIS Frontend

A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

%I A349314 #13 Nov 15 2021 08:56:40
%S A349314 1,2,18,274,4930,97346,2039570,44524818,1001773058,23065953794,
%T A349314 540886665618,12872727013522,310135678438978,7549240857128258,
%U A349314 185381380643501970,4586875745951650706,114244031335228433922,2862001783406012428802,72067481493990612275474
%N A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).
%C A349314 In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - _Vaclav Kotesovec_, Nov 15 2021
%H A349314 Seiichi Manyama, <a href="/A349314/b349314.txt">Table of n, a(n) for n = 0..500</a>
%F A349314 a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).
%F A349314 a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 14 2021
%F A349314 a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - _Vaclav Kotesovec_, Nov 15 2021
%t A349314 nmax = 18; A[_] = 0; Do[A[x_] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349314 Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
%Y A349314 Cf. A006318, A007556, A346626, A346650, A349310, A349311, A349312, A349313.
%K A349314 nonn
%O A349314 0,2
%A A349314 _Ilya Gutkovskiy_, Nov 14 2021