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%I #26 Jul 18 2023 10:31:57
%S 1,1,3,9,30,108,406,1577,6280,25499,105169,439388,1855636,7908909,
%T 33975250,146954693,639460707,2797384235,12295494109,54272825103,
%U 240480529815,1069257987503,4769306203838,21334400243252,95687482105807,430217846136134,1938651904470374,8754225470415889
%N G.f. satisfies A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)).
%C More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
%C F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
%C then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).
%H Alois P. Heinz, <a href="/A200074/b200074.txt">Table of n, a(n) for n = 0..450</a>
%F G.f. satisfies:
%F (1) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(n-k)] * x^n/n ).
%F (2) A(x) = exp( Sum_{n>=1} [(1-x/A(x))^(2*n+1)*Sum_{k>=0} C(n+k,k)^2*x^k/A(x)^k )] * x^n*A(x)^n/n.
%F (3) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A199876.
%F (4) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A199876.
%F Recurrence: (n+1)*(n+2)*(1241*n^4 - 10636*n^3 + 25417*n^2 - 7382*n - 17136)*a(n) = - 18*(n+1)*(443*n^3 - 3889*n^2 + 9734*n - 5712)*a(n-1) + 4*(6205*n^6 - 53180*n^5 + 115741*n^4 + 64762*n^3 - 370103*n^2 + 246727*n - 25704)*a(n-2) + 6*(2482*n^6 - 24995*n^5 + 76519*n^4 - 36347*n^3 - 185471*n^2 + 293092*n - 140400)*a(n-3) + 2*(4964*n^6 - 57436*n^5 + 228617*n^4 - 276802*n^3 - 361447*n^2 + 956696*n - 320496)*a(n-4) - 6*(2482*n^6 - 32441*n^5 + 140587*n^4 - 173153*n^3 - 266705*n^2 + 677518*n - 291840)*a(n-5) + 12*(n-4)*(2*n - 11)*(11*n^2 + 73*n - 748)*a(n-6) + 2*(n-5)*(2*n - 13)*(1241*n^4 - 5672*n^3 + 955*n^2 + 16508*n - 8496)*a(n-7). - _Vaclav Kotesovec_, Aug 18 2013
%F a(n) ~ c*d^n/n^(3/2), where d = 4.770539985405... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.612892860188927397373456... - _Vaclav Kotesovec_, Aug 18 2013
%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k+1,k) * binomial(2*n-3*k+1,n-2*k) / (2*n-3*k+1). - _Seiichi Manyama_, Jul 18 2023
%e G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 108*x^5 + 406*x^6 + 1577*x^7 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 87*x^4 + 330*x^5 + 1289*x^6 +...
%e A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 180*x^4 + 720*x^5 + 2928*x^6 +...
%e where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^3.
%e The logarithm of the g.f. A = A(x) equals the series:
%e log(A(x)) = (A + x)*x + (A^2 + 2^2*x*A + x^2)*x^2/2 +
%e (A^3 + 3^2*x*A^2 + 3^2*x^2*A + x^3)*x^3/3 +
%e (A^4 + 4^2*x*A^3 + 6^2*x^2*A^2 + 4^2*x^3*A + x^4)*x^4/4 +
%e (A^5 + 5^2*x*A^4 + 10^2*x^2*A^3 + 10^2*x^3*A^2 + 5^2*x^4*A + x^5)*x^5/5 +
%e (A^6 + 6^2*x*A^5 + 15^2*x^2*A^4 + 20^2*x^3*A^3 + 15^2*x^4*A^2 + 6^2*x^5*A + x^6)*x^6/6 +...
%e more explicitly,
%e log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 77*x^4/4 + 331*x^5/5 + 1445*x^6/6 + 6392*x^7/7 + 28565*x^8/8 +...
%p a:= n-> coeff(series(RootOf(A=(1+x*A^2)*(1+x^2*A), A), x, n+1), x, n):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 16 2012
%t m = 28; A[_] = 0;
%t Do[A[x_] = (1 + x A[x]^2)(1 + x^2 A[x]) + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 02 2019 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1+x*A^2)*(1+x^2*A^1)+x*O(x^n));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^j)*x^m*A^m/m))); polcoeff(A, n, x)}
%Y Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A004148, A036765.
%Y Cf. A186241, A199874, A200075, A200718, A200719, A215576.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 13 2011
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