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A118346 Central terms of pendular triangle A118345.

%I A118346 #19 Mar 17 2021 22:50:59
%S A118346 1,1,5,30,201,1445,10900,85128,682505,5585115,46461437,391743850,
%T A118346 3340361700,28755475180,249572076200,2181469638880,19186562661273,
%U A118346 169677521094215,1507881643936015,13458730170115778,120599648894147185
%N A118346 Central terms of pendular triangle A118345.
%C A118346 Also, g.f. A(x) = (1/x)*series_reversion of x/(1 + x*GF(A005572)), where GF(A005572) is the g.f. of A005572, which is the number of walks on cubic lattice starting and finishing on the xy plane and never going below it.
%H A118346 Seiichi Manyama, <a href="/A118346/b118346.txt">Table of n, a(n) for n = 0..1000</a>
%F A118346 G.f.: A=A(x) satisfies A = 1 - 2*x*A + 2*x*A^2 + x*A^3.
%F A118346 G.f.: A(x) = 1 + series_reversion( x/((1+x)*(1+4*x+x^2)) ).
%F A118346 G.f.: A(x) = (1/x)*series_reversion( x*(1-2*x + sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).
%F A118346 For n>0: a(n) = (1/n)*Sum_{j=0..n} Sum_{i=0..n-1} ( binomial(n,j) * binomial(j,i) * binomial(n-j,2*j-n-i-1) * 5^(2*n-3*j+2*i+1) ). -_Vladimir Kruchinin_, Dec 26 2010
%t A118346 CoefficientList[1 +InverseSeries[Series[x/((1+x)*(1+4*x+x^2)), {x,0,30}]], x] (* _G. C. Greubel_, Mar 17 2021 *)
%o A118346 (PARI) {a(n) = polcoeff(serreverse( x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x, n)}
%o A118346 (Sage)
%o A118346 def A118346_list(prec):
%o A118346     P.<x> = PowerSeriesRing(ZZ, prec)
%o A118346     return P( ( x/((1+x)*(1+4*x+x^2)) ).reverse() ).list()
%o A118346 a=A118346_list(31); [1]+a[1:] # _G. C. Greubel_, Mar 17 2021
%o A118346 (Magma)
%o A118346 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A118346 [1] cat Coefficients(R!( Reversion( x/((1+x)*(1+4*x+x^2)) ) )); // _G. C. Greubel_, Mar 17 2021
%Y A118346 Cf. A118345, A118347, A118348, A118349.
%K A118346 nonn
%O A118346 0,3
%A A118346 _Paul D. Hanna_, Apr 26 2006