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%I #8 Mar 19 2016 09:35:28
%S 1,1,1,1,2,4,1,3,12,28,1,4,24,114,276,1,5,40,288,1440,3480,1,6,60,580,
%T 4440,22368,53232,1,7,84,1020,10560,82080,409248,955524,1,8,112,1638,
%U 21420,226560,1752000,8585088,19672320,1,9,144,2464,38976,523320,5532960,42178800,202733760,456803328,1,10,180,3528,65520,1068480,14399280,150570240,1127335680,5317663680,11810032896,1,11,220,4860,103680,1991808,32716992,437433780,4501422240,33073099200,153345634560,336463895808
%N Rectangular table, read by antidiagonals, where the g.f. of row n, R(x,n), satisfies: R(x,n) = 1 + (n+1)*x*R(x,n+1)^2 for n>=0.
%C Row r > 0 is asymptotic to 2^(2*r) * n^r * A128318(n) / (3^r * r!). - _Vaclav Kotesovec_, Mar 19 2016
%H Paul D. Hanna, <a href="/A128570/b128570.txt">Table of n, a(n) for n = 0..527</a>
%e Row g.f.s satisfy: R(x,0) = 1 + x*R(x,1)^2, R(x,1) = 1 + 2x*R(x,2)^2,
%e R(x,2) = 1 + 3x*R(x,3)^2, R(x,3) = 1 + 4x*R(x,4)^2, ...
%e where the initial rows begin:
%e R(x,0):[1,1,4,28,276,3480,53232,955524,19672320,456803328,...];
%e R(x,1):[1,2,12,114,1440,22368,409248,8585088,202733760,...];
%e R(x,2):[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
%e R(x,3):[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
%e R(x,4):[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
%e R(x,5):[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
%e R(x,6):[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,..];
%e R(x,7):[1,8,144,3528,103680,3461760,127569600,5098406400,...];
%e R(x,8):[1,9,180,4860,156420,5690520,227470320,9821970180,...];
%e R(x,9):[1,10,220,6490,227040,8939040,385265760,17875608960,..].
%o (PARI) {T(n,k)=local(A=1+(n+k+1)*x); for(j=0,k,A=1+(n+k+1-j)*x*A^2 +x*O(x^k));polcoeff(A,k)}
%o for(n=0, 12, for(k=0, 10, print1(T(n, k), ", ")); print(""))
%Y Rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).
%Y Cf A268652.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Mar 11 2007
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