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A294360 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = (n^2)^(n-1) for n>=1.

%I A294360 #17 Mar 18 2024 07:20:19
%S A294360 1,1,5,146,9935,1161399,206499453,52093726159,17770811461875,
%T A294360 7903030237890371,4450363873663943294,3098938855124650814264,
%U A294360 2616552190721485829559668,2635178871851323631797948230,3121810359776427044817295874677,4298670834657263815567279951080956,6809336162211769799756516349665301635,12296952422064277377043754761717448273557,25116528778581121454413639996325045161219974
%N A294360 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = (n^2)^(n-1) for n>=1.
%H A294360 Paul D. Hanna, <a href="/A294360/b294360.txt">Table of n, a(n) for n = 0..200</a>
%F A294360 a(n) ~ c * n^(2*n - 2), where c = exp(2 - exp(-2)) = 6.453771681742981632532303... - _Vaclav Kotesovec_, Aug 11 2021, updated Mar 18 2024
%e A294360 G.f.: A(x) = 1 + x + 5*x^2 + 146*x^3 + 9935*x^4 + 1161399*x^5 + 206499453*x^6 + 52093726159*x^7 + 17770811461875*x^8 + 7903030237890371*x^9 + 4450363873663943294*x^10 + 3098938855124650814264*x^11 + 2616552190721485829559668*x^12 +...
%e A294360 such that the coefficient of x^(n-1) in A(x)^(n^2) equals (n^2)^(n-1) for n>=1.
%e A294360 The table of coefficients of x^k in A(x)^(n^2) begin:
%e A294360 n=1: [1, 1, 5, 146, 9935, 1161399, 206499453, ...];
%e A294360 n=2: [1, 4, 26, 648, 41703, 4775648, 840796570, ...];
%e A294360 n=3: [1, 9, 81, 1758, 102213, 11266209, 1949437539, ...];
%e A294360 n=4: [1, 16, 200, 4096, 207220, 21470032, 3617873616, ...];
%e A294360 n=5: [1, 25, 425, 8950, 390625, 36920005, 5985228975, ...];
%e A294360 n=6: [1, 36, 810, 18696, 723375, 60466176, 9272944890, ...];
%e A294360 n=7: [1, 49, 1421, 37338, 1347843, 97588547, 13841287201, ...];
%e A294360 n=8: [1, 64, 2336, 71168, 2535248, 159036480, 20303433408, 4398046511104, ...]; ...
%e A294360 in which the main diagonal begins:
%e A294360 [1, 4, 81, 4096, 390625, 60466176, 13841287201, 4398046511104, ..., (n^2)^(n-1), ...].
%o A294360 (PARI) {a(n) = my(A=[1]); for(m=2,n+1, A = concat(A,0); A[m] = ( (m^2)^(m-1) - Vec( Ser(A)^(m^2) )[m] )/m^2);A[n+1]}
%o A294360 for(n=0,20,print1(a(n),", "))
%Y A294360 Cf. A263190, A263075, A171791, A068087.
%K A294360 nonn
%O A294360 0,3
%A A294360 _Paul D. Hanna_, Nov 01 2017