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A233436 a(n) = Sum_{k=0..n-1} [x^k] A(x)^(n-1) for n>=1 with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.

%I A233436 #21 Sep 18 2024 05:14:44
%S A233436 1,1,2,8,50,424,4472,55760,797022,12801296,227829866,4446822688,
%T A233436 94422531876,2166975912496,53457972027254,1410960809766320,
%U A233436 39680975219789210,1184783226216138592,37434788449030871076,1248022160663960432264,43785432805297352937954,1612690422384099635004264
%N A233436 a(n) = Sum_{k=0..n-1} [x^k] A(x)^(n-1) for n>=1 with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%H A233436 Vaclav Kotesovec, <a href="/A233436/b233436.txt">Table of n, a(n) for n = 0..150</a>
%F A233436 Given g.f. A(x), let G(x) = A(x*G(x)), then A(x) = G(x/A(x)) = 1 + x*(G(x) + x*G'(x)) / (G(x) - x*G(x)^2).
%F A233436 a(n)/a(n-1) ~ n/LambertW(1). - _Vaclav Kotesovec_, Sep 14 2024
%e A233436 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 424*x^5 + 4472*x^6 + 55760*x^7 +...
%e A233436 ILLUSTRATION OF INITIAL TERMS.
%e A233436 If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
%e A233436 A^0 = [1],0,  0,   0,    0,     0,      0,       0,        0, ...;
%e A233436 A^1 = [1, 1], 2,   8,   50,   424,   4472,   55760,   797022, ...;
%e A233436 A^2 = [1, 2,  5], 20,  120,   980,  10056,  122960,  1732736, ...;
%e A233436 A^3 = [1, 3,  9,  37], 216,  1704,  17006,  203760,  2829030, ...;
%e A233436 A^4 = [1, 4, 14,  60,  345], 2640,  25632,  300744,  4111472, ...;
%e A233436 A^5 = [1, 5, 20,  90,  515,  3841], 36310,  417000,  5609960, ...;
%e A233436 A^6 = [1, 6, 27, 128,  735,  5370,  49493], 556212,  7359480, ...;
%e A233436 A^7 = [1, 7, 35, 175, 1015,  7301,  65723,  722765], 9400986, ...;
%e A233436 A^8 = [1, 8, 44, 232, 1366,  9720,  85644,  921864, 11782417], ...; ...
%e A233436 then a(n) equals the sum of the coefficients of x^k, k=0..n-1, in A(x)^(n-1) (shown above in brackets) for n>=1:
%e A233436 a(1) = 1 = 1;
%e A233436 a(2) = 1 +  1 = 2;
%e A233436 a(3) = 1 +  2 +  5 = 8;
%e A233436 a(4) = 1 +  3 +  9 +  37 = 50;
%e A233436 a(5) = 1 +  4 + 14 +  60 +  345 = 424;
%e A233436 a(6) = 1 +  5 + 20 +  90 +  515 + 3841 = 4472;
%e A233436 a(7) = 1 +  6 + 27 + 128 +  735 + 5370 + 49493 = 55760;
%e A233436 a(8) = 1 +  7 + 35 + 175 + 1015 + 7301 + 65723 + 722765 = 797022; ...
%e A233436 Also, from a diagonal in the above table we can obtain the coefficients:
%e A233436 [1/1, 2/2, 9/3, 60/4, 515/5, 5370/6, 65723/7, 921864/8, ...]
%e A233436 to form the power series
%e A233436 G(x) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 895*x^5 + 9389*x^6 + 115233*x^7 +...
%e A233436 that satisfies: A(x) = G(x/A(x)) = 1 + x*(G(x) + x*G'(x))/(G(x) - x*G(x)^2).
%o A233436 (PARI) {a(n)=local(A=1+x);if(n==0,1,for(i=1,n,
%o A233436 A=1+sum(k=1,n-1,sum(j=0,k-1,polcoeff(A^(k-1)+x*O(x^j),j))*x^k)+x*O(x^n));
%o A233436 sum(j=0,n-1,polcoeff(A^(n-1)+x*O(x^j),j)))}
%o A233436 for(n=0,20,print1(a(n),", "))
%Y A233436 Cf. A088713, A088358.
%K A233436 nonn
%O A233436 0,3
%A A233436 _Paul D. Hanna_, Dec 09 2013