A294360 - OEIS

%I #17 Mar 18 2024 07:20:19

%S 1,1,5,146,9935,1161399,206499453,52093726159,17770811461875,

%T 7903030237890371,4450363873663943294,3098938855124650814264,

%U 2616552190721485829559668,2635178871851323631797948230,3121810359776427044817295874677,4298670834657263815567279951080956,6809336162211769799756516349665301635,12296952422064277377043754761717448273557,25116528778581121454413639996325045161219974

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

%H Paul D. Hanna, <a href="/A294360/b294360.txt">Table of n, a(n) for n = 0..200</a>

%F 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 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 such that the coefficient of x^(n-1) in A(x)^(n^2) equals (n^2)^(n-1) for n>=1.

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

%e n=1: [1, 1, 5, 146, 9935, 1161399, 206499453, ...];

%e n=2: [1, 4, 26, 648, 41703, 4775648, 840796570, ...];

%e n=3: [1, 9, 81, 1758, 102213, 11266209, 1949437539, ...];

%e n=4: [1, 16, 200, 4096, 207220, 21470032, 3617873616, ...];

%e n=5: [1, 25, 425, 8950, 390625, 36920005, 5985228975, ...];

%e n=6: [1, 36, 810, 18696, 723375, 60466176, 9272944890, ...];

%e n=7: [1, 49, 1421, 37338, 1347843, 97588547, 13841287201, ...];

%e n=8: [1, 64, 2336, 71168, 2535248, 159036480, 20303433408, 4398046511104, ...]; ...

%e in which the main diagonal begins:

%e [1, 4, 81, 4096, 390625, 60466176, 13841287201, 4398046511104, ..., (n^2)^(n-1), ...].

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

%Y Cf. A263190, A263075, A171791, A068087.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 01 2017