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%I #14 Nov 05 2019 07:24:33
%S 1,5,85,2413,92501,4394663,246960721,15952488893,1161947365721,
%T 94123508334877,8390631582459161,816285612080072183,
%U 86069411025655759073,9778818588385117669485,1191176369495005591666205,154886342347657508336231809,21414816209632043592416524165,3137473307880710686085483679771,485584927860050612832028930482597,79169341280742628145184619086229089
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).
%H Paul D. Hanna, <a href="/A301435/b301435.txt">Table of n, a(n) for n = 0..50</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ( (1+x)^n - 3*x*A(x) )^n / 2^(n+1).
%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) / (2 + 3*x*A(x)*(1+x)^n)^(n+1). - _Paul D. Hanna_, Jan 10 2019
%F (3) 1 = Sum_{k>=0} (-3*x)^k * A(x)^k * Sum_{n>=0} C(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1).
%F (4) 1 = Sum_{n>=0} Sum_{k=0..n} C(n,k) * (1+x)^(n*(n-k)) / 2^(n+1) * (-3*x)^k * A(x)^k.
%e G.f.: A(x) = 1 + 5*x + 85*x^2 + 2413*x^3 + 92501*x^4 + 4394663*x^5 + 246960721*x^6 + 15952488893*x^7 + 1161947365721*x^8 + 94123508334877*x^9 + ...
%e such that
%e 1 = 1/2 + ((1+x) - 3*x*A(x))/2^2 + ((1+x)^2 - 3*x*A(x))^2/2^3 + ((1+x)^3 - 3*x*A(x))^3/2^4 + ((1+x)^4 - 3*x*A(x))^4/2^5 + ((1+x)^5 - 3*x*A(x))^5/2^6 + ...
%e Also,
%e 1 = 1/(2 + 3*x*A(x)) + (1+x)/(2 + 3*x*A(x)*(1+x))^2 + (1+x)^4/(2 + 3*x*A(x)*(1+x)^2)^3 + (1+x)^9/(2 + 3*x*A(x)*(1+x)^3)^4 + (1+x)^16/(2 + 3*x*A(x)*(1+x)^4)^5 + ...
%e RELATED SERIES.
%e Let R(k,x) = Sum_{n>=0} binomial(n+k,k) * (1+x)^(n*(n+k)) / 2^(n+k+1)
%e then
%e 1 = R(0,x) - 3*x*A(x)*R(1,x) + 3^2*x^2*A(x)^2*R(2,x) - 3^3*x^3*A(x)^3*R(3,x) + 3^4*x^4*A(x)^4*R(4,x) - 3^5*x^5*A(x)^5*R(5,x) + ...
%e The table of coefficients in R(k,x) begins:
%e k=0: [1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, ...];
%e k=1: [1, 10, 197, 5600, 206880, 9387864, 504836996, 31376330400, ...];
%e k=2: [1, 21, 621, 23447, 1078980, 58590504, 3667676768, ...];
%e k=3: [1, 36, 1494, 72516, 4075569, 261336096, 18861815280, ...];
%e k=4: [1, 55, 3050, 185190, 12492745, 934629539, 77091424200, ...];
%e k=5: [1, 78, 5571, 413764, 33004131, 2850142590, 266518090901, ...];
%e k=6: [1, 105, 9387, 837165, 77946645, 7696470411, 810015165897, ...];
%e k=7: [1, 136, 14876, 1568632, 168591350, 18874524760, 2221139481932, ...]; ...
%t nmax = 19;
%t nmax2 = 300 (* = empirical sum terms *);
%t sol = {a[0] -> 1};
%t A[x_] = Sum[a[n] x^n, {n, 0, nmax}];
%t Do[A[x] = A[x] /. sol; s = 1-Sum[((1+x)^n - 3x A[x] + O[x]^(k+1))^n / 2^(n+1), {n, 0, nmax2}] /. sol; c = SeriesCoefficient[s, {x, 0, k}]; sol = sol ~Join~ Solve[c == 0][[1]] /. HoldPattern[a[n_] -> an_] :> (a[n] -> Round[an]), {k, 2, nmax+1}];
%t a /@ Range[0, nmax] /. sol (* _Jean-François Alcover_, Nov 05 2019 *)
%Y Cf. A303288, A301436, A323314, A323315, A323316, A323317, A323318, A323319, A323320, A323321.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 24 2018