%I #27 Jun 28 2021 16:29:43
%S 1,1,2,6,22,90,393,1789,8378,40050,194492,956448,4752519,23822571,
%T 120315345,611644737,3127389106,16072642890,82981119252,430187414196,
%U 2238469102212,11687227631892,61208286479382,321465732705594
%N Revert transform of ((x - 1)(3x - 1))/(1 - 3x + x^3).
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F Revert transform of A122100 offset 1. - _Michael Somos_, May 11 2012
%F Recurrence: 12*(n-1)*n*(117*n - 604)*a(n) = 4*(n-1)*(4329*n^2 - 29719*n + 39384)*a(n-1) - 3*(23985*n^3 - 230056*n^2 + 679659*n - 630308)*a(n-2) + 3*(38961*n^3 - 437004*n^2 + 1609875*n - 1955812)*a(n-3) - 27*(n-4)*(2223*n^2 - 18496*n + 38133)*a(n-4) + 81*(n-5)*(n-4)*(117*n - 487)*a(n-5). - _Vaclav Kotesovec_, Jan 02 2021
%F a(n) ~ 3^(n - 1/4) * (2 + sqrt(3))^(n - 3/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - _Vaclav Kotesovec_, Jan 02 2021
%e G.f. = x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 90*x^6 + 393*x^7 + 1789*x^8 + ...
%t Rest[CoefficientList[InverseSeries[Series[x*((x - 1)(3x - 1))/(1 - 3x + x^3), {x, 0, 40}], x], x]] (* _Vaclav Kotesovec_, Jan 02 2021 *)
%o (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse(x * (1 - 4*x + 3*x^2) / (1 - 3*x + x^3) + x * O(x^n)), n))}; /* _Michael Somos_, May 11 2012 */
%Y Cf. A122100.
%K nonn
%O 1,3
%A _Olivier GĂ©rard_