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 A119365 Generalized Catalan numbers for triangle A119335. 5
 1, 0, 0, 1, 6, 20, 51, 126, 392, 1513, 5877, 21054, 71270, 242463, 863590, 3193737, 11889414, 43783908, 159998493, 586908936, 2175907284, 8138471667, 30541703733, 114620380032, 430344635913, 1619584557885, 6116422089050 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Counts rooted planar n-trees whose number of leaves is divisible by 3. LINKS Table of n, a(n) for n=0..26. FORMULA a(n) = A119335(2n,n) - A119335(2n,n+1). a(n) = Sum_{k=0..n} if(mod(n-k,3)=0, (1/n)*C(n,k)*C(n,k+1), 0). a(n) + A119366(n) + A119367(n) = A000108(n). MAPLE A119365 := proc(n) local k; if n = 0 then return 1 end if; a := 0 ; for k from 0 to n do if modp(n-k, 3) = 0 then a := a+binomial(n, k)*binomial(n, k+1) ; end if; end do: a/n; end proc: seq(A119365(n), n=0..40) ; # R. J. Mathar, Oct 30 2014 MATHEMATICA A119335[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}]; a[n_] := A119335[2n, n] - A119335[2n, n+1]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Sep 14 2023 *) CROSSREFS Cf. A000108, A001263, A119335, A119366, A119367. Sequence in context: A267168 A266760 A213586 * A001211 A122225 A275112 Adjacent sequences: A119362 A119363 A119364 * A119366 A119367 A119368 KEYWORD easy,nonn AUTHOR Paul Barry, May 16 2006 STATUS approved

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Last modified August 4 23:59 EDT 2024. Contains 374934 sequences. (Running on oeis4.)