Cher Simon,

  Voici la liste des suites de nombres a_n de 2-arbres k-gonaux
non-étiquetés ayant n k-gones, pour k allant de 2 à 12 et 
n=0...20. 
 
 La première est la suite  A000055 dans l'encyclopedia:
unlabelled trees. (k=2)

 La deuxième est A054581: unlabelled 2-trees (triangular) (k=3).
Les autres ne semblent pas y être.

 L'article a été accepté dans Journal of Combinatorial Theory,
series A mais tu peux aussi donner la référence des arXiv.

 Je t'envoie les constantes dans un autre message.

    Pierre Leroux




{

\begin{table}[h]

%\small%\scriptsize%
\footnotesize

{\boldmath$ k=2$}



1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 

48629, 123867, 317955, 823065, 2144505

%\hline



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=3$}



\noindent1, 1, 1, 2, 5, 12, 39, 136, 529, 2171, 9368, 41534, 188942, 874906, 

4115060, 19602156, 94419351, 459183768, 2252217207, 11130545494, 

55382155396



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=4$}



\noindent1, 1, 1, 3, 8, 32, 141, 749, 4304, 26492, 169263, 1115015, 7507211, 

51466500, 358100288, 2523472751, 17978488711, 129325796854, 

938234533024, 6858551493579, 50478955083341



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=5$}



\noindent1, 1, 1, 3, 11, 56, 359, 2597, 20386, 167819, 1429815, 12500748, 

111595289, 1013544057, 9340950309, 87176935700, 822559721606, 

7836316493485, 75293711520236, 728968295958626, 7105984356424859



%\noindent--------------------------------------------------------



\noindent{\boldmath$  k=6$}



1, 1, 1, 4, 16, 103, 799, 7286, 71094, 729974, 7743818, 84307887, 

937002302, 10595117272, 121568251909, 1412555701804, 16594126114458, 

196829590326284, 2354703777373055, 28385225424840078, 

344524656398655124



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=7$}



\noindent1, 1, 1, 4, 20, 158, 1539, 16970, 199879, 2460350, 31266165, 

407461893, 5420228329, 73352481577, 1007312969202, 14008437540003, 

196963172193733, 2796235114720116, 40038505601111596, 

577693117173844307, 8392528734991449808



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=8$}



\noindent1, 1, 1, 5, 26, 245, 2737, 35291, 483819, 6937913, 102666626, 

1558022255, 24133790815, 380320794122, 6081804068869, 98490990290897, 

1612634990857755, 26660840123167203, 444560998431678554, 

7469779489114328514, 126375763235359105446



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=9$}



1, 1, 1, 5, 32, 343, 4505, 66603, 1045335, 17115162, 289107854, 

5007144433, 88516438360, 1591949961503, 29053438148676, 

536972307386326, 10034276171127780, 189331187319203010, 

3603141751525175854, 69097496637591215442, 1334213677527481808220



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=10$}



1, 1, 1, 6, 39, 482, 7053, 117399, 2070289, 38097139, 723169329, 

14074851642, 279609377638, 5651139037570, 115901006038377, 

2407291353219949, 50553753543016719, 1071971262516091572, 

22926544048209731554, 494103705426160765546, 10722146465907412669810



%\noindent--------------------------------------------------------



\noindent{\boldmath$ k=11$}



\noindent1, 1, 1, 6, 46, 636, 10527, 194997, 3823327, 78118107, 1646300388, 

35570427615, 784467060622, 17601062294302, 400750115756742, 

9240636709048733, 215435023547580882, 5071520482516388865, 

120417032326341878672, 2881134828445365441407, 69410468220307148620226



%\noindent--------------------------------------------------------

%\noindent--------------------------------------------------------



\noindent{\boldmath$  k=12$}



\noindent1, 1, 1, 7, 55, 840, 15189, 309607, 6671842, 149850849, 3471296793, 

82442359291, 1998559329142, 49290785442796, 1233639304644946, 

31268489727956101, 801335133177932829, 20736286803363051714, 

541224489038545084067, 14234799536039481373552, 

376974819516101224941091

%

%\noindent--------------------------------------------------------

%}

\caption{Values of $\widetilde{a}_{n}$ for $k=2,\ldots,12$ and $n=0,\ldots,20$}

\label{tableantilde}

\end{table}

}