This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381359 #50 Apr 02 2025 05:16:11
%S A381359 1,12,720,129600,51321600,37977984000,47113228800000,
%T A381359 90796614543360000,256892229695692800000,1021474451008342425600000,
%U A381359 5513370502054734544896000000,39267642006336798923489280000000,360478517037545726209161953280000000,4181620210850033164370074219315200000000
%N A381359 E.g.f. A(x) satisfies 1 - A'(x)^2 + 4*A(x)^3 = 0.
%C A381359 E.g.f. A(x) equals a series trisection of the e.g.f. F(x) of A381360, which satisfies 1 + 3*F'(x)^2 - 4*F(x)^3 = 0. See A381360 for more formulas in which the trisection T1 denotes the e.g.f. A(x) of this sequence.
%H A381359 Paul D. Hanna, <a href="/A381359/b381359.txt">Table of n, a(n) for n = 0..200</a>
%H A381359 Roland Bacher and Philippe Flajolet, <a href="http://arxiv.org/abs/0901.1379">Pseudo-factorials, elliptic functions, and continued fractions</a>, arXiv:0901.1379 [math.CA], 2009.
%H A381359 Eric van Fossen Conrad and Philippe Flajolet, <a href="http://www.emis.de/journals/SLC/wpapers/s54conflaj.html">The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion</a>, Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.
%F A381359 E.g.f. A(x) = Sum_{n>=0} a(n)*x^(3*n+1)/(3*n+1)! satisfies the following formulas.
%F A381359 (1) A'(x) = sqrt(1 + 4*A(x)^3).
%F A381359 (2) A'(x) = 1 + 2*Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^3.
%F A381359 (3.a) A'(x) = -1 - 2*A(x)/A(-x).
%F A381359 (3.b) A'(x) = -1 + 2*exp( 3*Integral -A(x)*A(-x) dx ).
%F A381359 (4.a) A''(x) = 6*A(x)^2.
%F A381359 (4.b) A'''(x) = 12*A(x)*A'(x).
%F A381359 (4.c) A''''(x) = 12 + 120*A(x)^3.
%F A381359 (5) A(x) = Series_Reversion( Integral 1/sqrt(1 + 4*x^3) dx ).
%F A381359 (6) A(x) = x + 2*Integral Series_Reversion( Integral 1/(1 + x^3)^(2/3) dx )^3 dx.
%F A381359 (7) A(x) = x + Integral Integral 6*A(x)^2 dx dx.
%F A381359 (8.a) A(x) = -x + Integral -2*A(x)/A(-x) dx.
%F A381359 (8.b) A(x) = -A(-x) * exp( 3*Integral -A(x)*A(-x) dx ).
%F A381359 (8.c) A(x) = x * exp( Integral -2/A(-x) - 1/A(x) - 1/x dx ).
%F A381359 (8.d) A(x) = x * exp( Integral sqrt(1 + 4*A(x)^3)/A(x) - 1/x dx ).
%F A381359 (9.a) A(x) = -cm(-x)*sm(-x) where cm(x) and sm(x) are Dixon elliptic functions (A104134 and A104133) satisfying cm(x)^3 + sm(x)^3 = 1.
%F A381359 (9.b) cm(-x) = exp( (1/2)*Integral (A'(x) - 1)/A(x) dx ).
%F A381359 (9.c) sm(-x) = -x*exp( (1/2)*Integral (A'(x) + 1)/A(x) - 2/x dx ).
%F A381359 From _Thomas Scheuerle_ and _Paul D. Hanna_, Mar 07 2025: (Start)
%F A381359 O.g.f.: 1/T(1, x), where T(k, x) = (1 - 3*k*(3*k-1)^2*x/(1 - 3*k*(3*k+1)^2*x/T(k+1, x))) (continued fraction).
%F A381359 O.g.f.: 1/T(1, x), where T(k, x) = (1 - 3*(2*k - 1)*(9*k^2 - 9*k + 4)*x - ((3*k + 1)*(3*k)*(3*k - 1))^2*x^2/T(k+1, x)) (continued fraction). (End)
%e A381359 E.g.f.: A(x) = x + 12*x^4/4! + 720*x^7/7! + 129600*x^10/10! + 51321600*x^13/13! + 37977984000*x^16/16! + ... + a(n)*x^(3*n+1)/(3*n+1)! + ... where A'(x) = sqrt(1 + 4*A(x)^3).
%e A381359 The ordinary generating function begins
%e A381359 O.g.f.: G(x) = 1 + 12*x + 720*x^2 + 129600*x^3 + 51321600*x^4 + 37977984000*x^5 + 47113228800000*x^6 + ...
%e A381359 Continued fraction representations for the o.g.f. are as follows.
%e A381359 O.g.f.: G(x) = 1/(1 - 12*x/(1 - 48*x/(1 - 150*x/(1 - 294*x/(1 - 576*x/(1 - 900*x/(1 - ...))))))).
%e A381359 O.g.f.: G(x) = 1/(1-12*x - 24^2*x^2/(1-198*x - 210^2*x^2/(1-870*x - 720^2*x^2/(1-2352*x - 1716^2*x^2/(1-4968*x - ...))))).
%o A381359 (PARI) {a(n) = my(A = serreverse( intformal( 1/sqrt(1 + 4*x^3 + x*O(x^(3*n+1))) ) )); (3*n+1)!*polcoef(A,3*n+1)}
%o A381359 for(n=0,20, print1(a(n),", "))
%Y A381359 Cf. A381360, A104133, A104134.
%K A381359 nonn
%O A381359 0,2
%A A381359 _Paul D. Hanna_, Mar 06 2025