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%I #29 May 22 2023 02:14:54
%S 1,3,11,42,165,671,2795,11877,51286,224413,992924,4434833,19969030,
%T 90550829,413148619,1895338362,8737219074,40452543831,188025758635,
%U 877055405522,4104269624748,19262955163275,90652992751518,427681283728070,2022341915324936,9583224591208298
%N a(n) = coefficient of x^n in A(x) such that 1 = Sum_{n=-oo..+oo} (-x)^n * (A(x) + x^(n-1))^(n+1).
%C Row sums of triangle A359670.
%H Paul D. Hanna, <a href="/A359711/b359711.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F (1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(n-1))^(n+1).
%F (2) x = Sum_{n=-oo..+oo} (-1)^n * (x*A(x) + x^n)^(n+1).
%F (3) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+1))^(n-1).
%F (4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (x*A(x) + x^n)^n ].
%F (5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^(n+1))^n ].
%F From _Paul D. Hanna_, May 18 2023: (Start)
%F (6) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x) + x^n)^n.
%F (7) A(x) = -1 / [Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^n ].
%F (8) x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A*x^(n+1))^(n+1).
%F (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^n)^(n+1).
%F (10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^n)^n. (End)
%F a(n) = Sum_{k=0..n} A359670(n,k) for n >= 0.
%F a(n) ~ c * d^n / n^(3/2), where d = 5.008723344615566939692217... and c = 4.45330627132612826203... - _Vaclav Kotesovec_, Mar 14 2023
%e G.f.: A(x) = 1 + 3*x + 11*x^2 + 42*x^3 + 165*x^4 + 671*x^5 + 2795*x^6 + 11877*x^7 + 51286*x^8 + 224413*x^9 + 992924*x^10 + ...
%o (PARI) {a(n) = my(A=1,y=1); for(i=1,n,
%o A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
%o polcoeff( A,n,x)}
%o for(n=0,25, print1( a(n),", "))
%o (PARI) {a(n) = my(A=[1],y=1); for(i=1,n, A = concat(A,0);
%o A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
%o for(n=0,25, print1( a(n),", "))
%Y Cf. A359670, A359712, A359713, A363104, A363105.
%Y Cf. A363142, A363143, A363144.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 17 2023
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