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A179848 Expansion of series reversion of generating function for triangular numbers.

%I A179848 #20 Mar 24 2023 14:34:06
%S A179848 0,1,-3,12,-55,273,-1428,7752,-43263,246675,-1430715,8414640,
%T A179848 -50067108,300830572,-1822766520,11124755664,-68328754959,
%U A179848 422030545335,-2619631042665,16332922290300,-102240109897695,642312451217745,-4048514844039120,25594403741131680
%N A179848 Expansion of series reversion of generating function for triangular numbers.
%H A179848 Alois P. Heinz, <a href="/A179848/b179848.txt">Table of n, a(n) for n = 0..375</a>
%F A179848 A001764(n) = 0^n - (-1)^n * a(n).
%F A179848 G.f. A(x) satisfies A(x) = x * (1 - A(x))^3.
%F A179848 G.f.: 1 - sinh( arcsinh( sqrt( 27*x/4 ) ) / 3 ) / sqrt( 3*x/4 ).
%F A179848 D-finite with recurrence +2*n*(2*n+1)*a(n) +3*(3*n-1)*(3*n-2)*a(n-1)=0. - _R. J. Mathar_, Mar 24 2023
%e A179848 G.f. = x - 3*x^2 + 12*x^3 - 55*x^4 + 273*x^5 - 1428*x^6 + 7752*x^7 - 43263*x^8 + ...
%p A179848 a:= n-> coeff(series(RootOf(A=x*(1-A)^3, A), x, n+1), x, n):
%p A179848 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 16 2012
%p A179848 # Using function CompInv from A357588.
%p A179848 0, CompInv(23, n -> n*(n+1)/2); # _Peter Luschny_, Oct 05 2022
%t A179848 CoefficientList[Series[1 - Sinh[ArcSinh[Sqrt[27*x/4]]/3]/Sqrt[3*x/4], {x, 0, 50}], x] (* _G. C. Greubel_, Aug 14 2018 *)
%o A179848 (PARI) {a(n) = if( n<1, 0, -(-1)^n * (3*n)! / (n! * (2*n+1)!) )};
%o A179848 (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x / (1 - x)^3 + x * O(x^n) ), n))};
%o A179848 (PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for( k = 0, n, A = x * (1 - A)^3 ); polcoeff( A, n ))};
%o A179848 (Magma) [n le 0 select 0 else (-1)^(n+1)*Factorial(3*n)/( Factorial(n)* Factorial(2*n+1)): n in [0..30]]; // _G. C. Greubel_, Aug 14 2018
%Y A179848 Cf. A000217.
%K A179848 sign
%O A179848 0,3
%A A179848 _Michael Somos_, Jan 10 2011