A349303 - OEIS

%I #14 Jun 06 2024 07:14:25

%S 1,0,1,7,57,483,4257,38675,359969,3416329,32943289,321888455,

%T 3180249409,31718822793,318934721393,3229639622847,32907617157641,

%U 337144842511850,3470986886039193,35890957497118363,372584381500477185,3881595191885835547

%N G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^7)).

%C In general, for k>=1, Sum_{j=0..n} (-1)^(n-k) * binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 - (k-1)*r) / (sqrt(2*k*(k+1)*(1+r)*Pi) * (k+1)^(1/k) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1+r)^k. - _Vaclav Kotesovec_, Nov 14 2021

%H Seiichi Manyama, <a href="/A349303/b349303.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).

%F a(n) ~ sqrt(1 - 6*r) / (2^(17/7) * sqrt(7*Pi*(1+r)) * n^(3/2) * r^(n + 1/7)), where r = 0.08937121041965233233945479666512758370169477786851479485467... is the real root of the equation 8^8 * r = 7^7 * (1+r)^7. - _Vaclav Kotesovec_, Nov 14 2021

%F From _Peter Bala_, Jun 02 2024: (Start)

%F A(x) = 1/(1 + x)*F(x/(1 + x)^7), where F(x) = Sum_{n >= 0} A007556(n)*x^n.

%F A(x) = 1/(1 + x) + x*A(x)^8. (End)

%t nmax = 21; A[_] = 0; Do[A[x_] = 1/((1 + x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[Sum[(-1)^(n - k) Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 21}]

%Y Cf. A005043, A007556, A346627, A346668, A349293, A349299, A349300, A349301, A349302.

%K nonn,easy

%O 0,4

%A _Ilya Gutkovskiy_, Nov 13 2021