A307402 - OEIS

%I #7 Apr 07 2019 08:35:54

%S 1,1,4,23,164,1362,12792,133891,1550148,19772030,277054232,4252637446,

%T 71248226536,1297226168708,25542157054944,541131735552507,

%U 12275049552454916,296787898215881990,7617196890240489912,206772478080888288082,5917589117194665548600,178040033221054576103036

%N G.f. A(x) satisfies: A(x) = Sum_{j>=0} j!*x^j*A(x)^j / Product_{k=1..j} (1 - k*x*A(x)).

%H Vaclav Kotesovec, <a href="/A307402/b307402.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) satisfies: A(x) = Sum_{k>=0} A000670(k)*x^k*A(x)^k.

%F G.f.: A(x) = (1/x)*Series_Reversion(x/Sum_{k>=0} A000670(k)*x^k).

%F a(n) ~ n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Apr 07 2019

%e G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 164*x^4 + 1362*x^5 + 12792*x^6 + 133891*x^7 + 1550148*x^8 + 19772030*x^9 + 277054232*x^10 + ...

%t terms = 22; A[_] = 1; Do[A[x_] = Sum[j! x^j A[x]^j/Product[(1 - k x A[x]), {k, 1, j}], {j, 0, i}] + O[x]^i, {i, 1, terms}]; CoefficientList[A[x], x]

%t terms = 22; A[_] = 1; Do[A[x_] = Sum[(1/2) HurwitzLerchPhi[1/2, -k, 0] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%t terms = 22; CoefficientList[1/x InverseSeries[Series[x/Sum[(1/2) HurwitzLerchPhi[1/2, -k, 0]  x^k, {k, 0, terms}], {x, 0, terms}], x], x]

%Y Cf. A000670, A224922.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 07 2019