A049126 Revert transform of ((x - 1)(3x - 1))/(1 - 3x + x^3).

1, 1, 2, 6, 22, 90, 393, 1789, 8378, 40050, 194492, 956448, 4752519, 23822571, 120315345, 611644737, 3127389106, 16072642890, 82981119252, 430187414196, 2238469102212, 11687227631892, 61208286479382, 321465732705594

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Examples

			G.f. = x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 90*x^6 + 393*x^7 + 1789*x^8 + ...
		

Links

• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Index entries for reversions of series

Crossrefs

Cf. A122100.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x*((x - 1)(3x - 1))/(1 - 3x + x^3), {x, 0, 40}], x], x]] (* Vaclav Kotesovec, Jan 02 2021 *)

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 */

Formula

Revert transform of A122100 offset 1. - Michael Somos, May 11 2012

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

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