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A103779 Series reversion of y + y^2 + y^3.

%I A103779 #67 Apr 06 2025 19:53:06
%S A103779 0,1,-1,1,0,-4,14,-30,33,55,-429,1365,-2652,1428,12920,-64600,178296,
%T A103779 -277932,-152950,2770350,-10785390,25312650,-26053020,-84847620,
%U A103779 576753450,-1856900682,3566658438,-843350102,-24973594296,117328602840,-317641049880,455822225496
%N A103779 Series reversion of y + y^2 + y^3.
%C A103779 Second column of A103778 (inverse of trinomial triangle A071675).
%H A103779 R. J. Mathar, <a href="/A103779/b103779.txt">Table of n, a(n) for n = 0..103</a>
%H A103779 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F A103779 G.f.: -2^(2/3) * ((3*sqrt(3)*sqrt(27*x^2+14*x+3)-27*x-7)^(1/3) -(3*sqrt(3) * sqrt(27*x^2+14*x+3)+27*x+7)^(1/3) +2^(1/3))/6.
%F A103779 a(0) = 0, a(1) = 1, and, for n >= 2, a(n) = -Sum_{k = 1..n-1} a(k) * Sum_{j = 0..k} C(j,n-k-j) * C(k,j). - _Vladimir Kruchinin_, Apr 08 2011
%F A103779 a(n) = (1/n)*Sum_{k = 1..n-1} (-1)^k * C(k,n-1-k) * C(n+k-1,n-1), with a(1) = 1. - _Vladimir Kruchinin_, May 12 2012
%F A103779 D-finite with recurrence 3*n*(n-1)*a(n) +7*(n-1)*(2*n-3)*a(n-1) +3*(3*n-5)*(3*n-7)*a(n-2)=0. - _R. J. Mathar_, Oct 06 2012
%F A103779 G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ). - _Paul D. Hanna_, Apr 17 2016
%F A103779 From _Paul D. Hanna_, Sep 06 2022: (Start)
%F A103779 G.f. A(x) satisfies:
%F A103779 A(x)^5 = A( x^5 - 5*x*(1+x)^2*A(x)^5 ), and
%F A103779 A(x)^5 = ( x^5 - 5*x*(1+x)^2*A(x)^5 ) * (1 - A(x)^5) / (1 - A(x)^15). (End)
%F A103779 A(x) = x*exp(Sum_{n >= 1} A350383(n)*x^n/n). - _Peter Bala_, Mar 11 2025
%e A103779 G.f.: A(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + 33*x^8 + 55*x^9 - 429*x^10 + 1365*x^11 - 2652*x^12 + 1428*x^13 + 12920*x^14 + ... where A(x + x^2 + x^3) = x.
%t A103779 CoefficientList[ InverseSeries[ Series[y + y^2 + y^3, {y, 0, 28}], x], x] (* _Robert G. Wilson v_ *)
%o A103779 (Maxima) a(n):=if n=1 then 1 else -sum(sum(binomial(j,n-k-j) *binomial(k,j),j,0,k)*a(k),k,1,n-1); /* _Vladimir Kruchinin_, Apr 08 2011 */
%o A103779 (Maxima) a(n):=if n=1 then 1 else 1/n*sum(binomial(k,n-1-k)*(-1)^k *binomial(n+k-1,n-1),k,1,n-1); /* _Vladimir Kruchinin_, May 12 2012 */
%o A103779 (PARI) Vec(serreverse(x*(1+x+x^2)+O(x^66))) /* _Joerg Arndt_, Aug 19 2012 */
%o A103779 (PARI) /* G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ) */
%o A103779 {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2 - 2*X*A^2)^(1/2) ); polcoeff(A, n)}
%o A103779 for(n=1, 40, print1(a(n), ", ")) \\ _Paul D. Hanna_, Apr 17 2016
%Y A103779 Cf. A350383.
%K A103779 easy,sign
%O A103779 0,6
%A A103779 _Paul Barry_, Feb 15 2005