A378325 - OEIS

A378325

G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.

%I #13 Nov 24 2024 06:15:48

%S 1,1,2,7,41,338,3499,42969,606351,9633640,169888025,3290314970,

%T 69409429043,1584105116525,38894316619948,1022411500472240,

%U 28653072049382809,852911635849385778,26876978490909421289,893929164892155754432,31296785296935394097351,1150551256823546563078988

%N G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.

%H Vaclav Kotesovec, <a href="/A378325/b378325.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * n! / (n^alpha * LambertW(1)^n), where alpha = 2 - 2*LambertW(1) - 1/(1 + LambertW(1)) = 0.22760967581532... and c = 0.323194722450152336...

%o (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);

%o A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^k, k)) )); A[n+1]}

%o for(n=0, 30, print1(a(n), ", ")) \\ after _Paul D. Hanna_

%Y Cf. A377098, A378326.

%Y Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Nov 23 2024