%I #8 Feb 03 2018 15:42:30

%S 1,1,5,90,3204,170987,12162683,1087504130,118227836360,15304211345298,

%T 2324856843115770,409872125913866852,83092182794794380856,

%U 19214014336799266619671,5030971580159960051721815,1481724835890098667273954338,487883202104697456579537247232,178595806151469762148235569612814,72312528698655521190143801630975174

%N G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2 with A'(0) = 1.

%C Compare g.f. to: [x^(n-1)] G(x)^(n^2)/n^2 = [x^(n-2)] G(x)^(n^2)/(n-1) for n>=2 holds when G(x) = exp(x).

%H Paul D. Hanna, <a href="/A295766/b295766.txt">Table of n, a(n) for n = 0..260</a>

%F a(A075427(k) - 1) is odd for n>=0 and a(n) is even elsewhere (conjecture).

%e G.f.: A(x) = 1 + x + 5*x^2 + 90*x^3 + 3204*x^4 + 170987*x^5 + 12162683*x^6 + 1087504130*x^7 + 118227836360*x^8 + 15304211345298*x^9 + 2324856843115770*x^10 + ...

%e ILLUSTRATION OF THE DEFINITION.

%e The table of coefficients of x^k in A(x)^(n^2) begins:

%e n=1: [1, 1, 5, 90, 3204, 170987, 12162683, ...];

%e n=2: [1, 4, 26, 424, 14107, 729196, 50993674, ...];

%e n=3: [1, 9, 81, 1254, 37602, 1833597, 124332453, ...];

%e n=4: [1, 16, 200, 3200, 86084, 3846720, 248466736, ...];

%e n=5: [1, 25, 425, 7550, 188750, 7566705, 455263225, ...];

%e n=6: [1, 36, 810, 16680, 410499, 14777964, 808802730, ...];

%e n=7: [1, 49, 1421, 34594, 886312, 29473255, 1444189495, ...]; ...

%e in which the main diagonal

%e [1, 4, 81, 3200, 188750, 14777964, 1444189495, ...]

%e is related to an adjacent diagonal by dividing by n^2 like so:

%e [1, 4/4, 81/9, 3200/16, 188750/25, 14777964/36, 1444189495/49, ...]

%e = [1, 1, 9, 200, 7550, 410499, 29473255, ...].

%e Thus [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2.

%o (PARI) {a(n) = my(A=[1],V); for(m=2,n+1, A=concat(A,0); V=Vec(Ser(A)^(m^2)); A[#A] = V[#A-1] - V[#A]/m^2 );A[n+1]}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) /* Informal method of obtaining N terms: */

%o N=30; A=[1]; for(n=2,N, A=concat(A,0); V=Vec(Ser(A)^(n^2)); A[#A] = V[#A-1] - V[#A]/n^2 );A

%Y Cf. A088715, A295811, A075427.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 31 2018