%I #9 Mar 30 2012 18:37:35

%S 1,1,3,22,285,5476,143080,4840256,205348521,10675415800,667749546288,

%T 49499829165414,4292944063388838,430749877491962328,

%U 49520529524763502478,6467220964216618610444,952209872438113961762005,157000879970348439087083080

%N a(n) equals the sum of the coefficients of x^(n-1) in A(x)^k as k=1..n, where A(x) = Sum_{n>=0} a(n)*x^n.

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 285*x^4 + 5476*x^5 + 143080*x^6 +...

%e The coefficients in the powers of the g.f. A(x) begin:

%e A^1: [1, 1, 3, 22, 285, 5476, 143080, 4840256, ...];

%e A^2: [1, 2, 7, 50, 623, 11654, 299306, 10012068, ...];

%e A^3: [1, 3, 12, 85, 1023, 18627, 469956, 15539040, ...];

%e A^4: [1, 4, 18, 128, 1495, 26500, 656450, 21446816, ...];

%e A^5: [1, 5, 25, 180, 2050, 35391, 860365, 27763280, ...];

%e A^6: [1, 6, 33, 242, 2700, 45432, 1083451, 34518774, ...];

%e A^7: [1, 7, 42, 315, 3458, 56770, 1327648, 41746335, ...];

%e A^8: [1, 8, 52, 400, 4338, 69568, 1595104, 49481952, ...]; ...

%e where a(n) = sum of the first n coefficients in column n-1 for n>=1:

%e a(1) = 1,

%e a(2) = 1 + 2 = 3;

%e a(3) = 3 + 7 + 12 = 22;

%e a(4) = 22 + 50 + 85 + 128 = 285;

%e a(5) = 285 + 623 + 1023 + 1495 + 2050 = 5476; ...

%o (PARI) {a(n)=local(A=[1]);for(i=1,n,A=concat(A,sum(m=1,#A,Vec(Ser(A)^m)[#A])));A[n+1]}

%o for(n=0,21,print1(a(n),", "))

%Y Cf. A088358.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 12 2012