|
%I #23 Jun 18 2022 14:37:41
%S 0,1,2,6,19,63,216,758,2705,9777,35698,131425,487201,1816651,6807742,
%T 25621878,96796225,366902949,1394851446,5316835073,20314772302,
%U 77786795230,298435201100,1147019162326,4415737088310,17025146600174,65732992038182,254118443847070,983579262641569
%N Total number of left-to-right maxima in Dyck paths of semilength n.
%H Aubrey Blecher and Arnold Knopfmacher, <a href="https://arxiv.org/abs/2107.03102">Left to right maxima in Dyck Paths</a>, arXiv:2107.03102 [math.CO], 2021. See Theorem 4 p. 8.
%F a(n) = Sum_{r=1..n} (d(r+1)-d(r))*(binomial(2*n-1, n-r)-binomial(2*n-1,n-r-1)) where d is A000005 and binomial is A007318.
%p a:= n-> (d-> add((binomial(2*n-1, n-r)-binomial(2*n-1, n-r-1))
%p *(d(r+1)-d(r)), r=1..n))(numtheory[tau]):
%p seq(a(n), n=0..28); # _Alois P. Heinz_, Jul 08 2021
%t a[n_] := (Sum[(DivisorSigma[0, r + 1] - DivisorSigma[0, r])*(Binomial[2*n - 1, n - r] - Binomial[2*n - 1, n - r - 1]), {r, 1, n}]);
%t Table[a[i], {i, 0, 28}] (* _Kebbaj Mohamed Reda_, Jun 06 2022 *)
%o (PARI) a(n) = sum(r=1, n, (numdiv(r+1)-numdiv(r))*(binomial(2*n-1, n-r)-binomial(2*n-1,n-r-1)));
%Y Cf. A000005, A007318, A346158, A346194.
%K nonn
%O 0,3
%A _Michel Marcus_, Jul 08 2021
|
