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%I #9 Aug 11 2021 13:45:04
%S 1,1,4,63,2116,113550,8687239,891701755,118110608484,19625931563166,
%T 4001492423904134,983010151440620919,286534726440573778723,
%U 97810529840988133377195,38660405402434292659485223,17520204438205012495544783268
%N G.f. satisfies: a(n) = [x^n] ( A(x)^n + 1/A(x)^n ) for n>1, with a(0)=a(1)=1.
%H Paul D. Hanna, <a href="/A237644/b237644.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ c * 2^n * n^(2*n + 1) / exp(2*n), where c = 2.15155017267809206855547... - _Vaclav Kotesovec_, Aug 11 2021
%e G.f.: A(x) = 1 + x + 4*x^2 + 63*x^3 + 2116*x^4 + 113550*x^5 + 8687239*x^6 +...
%e Given the table of coefficients of x^k in A(x)^n, for {-oo < n < oo}:
%e ...
%e [1,-7, 0,-301,-11634, -679252,-54467609,-5754393863, ...];
%e [1,-6, -3,-266,-10260, -593352,-47319971,-4982057256, ...];
%e [1,-5, -5,-230, -8800, -504011,-39971530,-4193722200, ...];
%e [1,-4, -6,-192, -7249, -411076,-32416322,-3389053044, ...];
%e [1,-3, -6,-151, -5601, -314385,-24648147,-2567705130, ...];
%e [1,-2, -5,-106, -3849, -213766,-16660556,-1729324458, ...];
%e [1,-1, -3, -56, -1985, -109036, -8446837, -873547334, ...];
%e [1, 0, 0, 0, 0, 0, 0, 0, ...];
%e [1, 1, 4, 63, 2116, 113550, 8687239, 891701755, ...];
%e [1, 2, 9, 134, 4374, 231836, 17622475, 1801953004, ...];
%e [1, 3, 15, 214, 6786, 355095, 26813632, 2731160112, ...];
%e [1, 4, 22, 304, 9365, 483580, 36268982, 3679741188, ...];
%e [1, 5, 30, 405, 12125, 617561, 45997165, 4648126560, ...];
%e [1, 6, 39, 518, 15081, 757326, 56007210, 5636759274, ...];
%e [1, 7, 49, 644, 18249, 903182, 66308557, 6646095618, ...];
%e ...
%e then the table of coefficients of x^k in A(x)^n + 1/A(x)^n begins:
%e [2, 0, 0, 0, 0, 0, 0, 0, ...];
%e [2, 0, 1, 7, 131, 4514, 240402, 18154421, ...];
%e [2, 0, 4, 28, 525, 18070, 961919, 72628546, ...];
%e [2, 0, 9, 63, 1185, 40710, 2165485, 163454982, ...];
%e [2, 0, 16, 112, 2116, 72504, 3852660, 290688144, ...];
%e [2, 0, 25, 175, 3325, 113550, 6025635, 454404360, ...];
%e [2, 0, 36, 252, 4821, 163974, 8687239, 654702018, ...];
%e [2, 0, 49, 343, 6615, 223930, 11840948, 891701755, ...];
%e ...
%e such that the main diagonal yields this sequence for n>1.
%o (PARI) {a(n)=local(A=vector(n+1,i,1)); for(n=2,#A-1, A[n+1] = Vec(Ser(A)^n + 1/Ser(A)^n)[n+1]); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* Build vector of n=0..30 terms (faster): */
%o {A=vector(31,i,1); for(n=2,#A-1, A[n+1] = Vec(Ser(A)^n + 1/Ser(A)^n)[n+1]); A}
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 02 2014
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