

A121523


Number of up steps starting at an even level in all nondecreasing Dyck paths of semilength n. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.


2



1, 3, 10, 33, 103, 315, 941, 2770, 8051, 23171, 66138, 187486, 528365, 1481501, 4135756, 11500721, 31871625, 88054825, 242609585, 666783380, 1828452021, 5003697403, 13667302500, 37267071708, 101455834153, 275797332135
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OFFSET

1,2


COMMENTS

a(n) = Sum(k*A121522(n,k), k=1..n). a(n)+A121525(n)=n*fibonacci(2n1).


LINKS



FORMULA

G.f.: z(13z+z^2+5z^35z^4)/[(1+z)(13z+z^2)^2*(1zz^2)].
a(n) ~ (5sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+2)).  Vaclav Kotesovec, Mar 20 2014


EXAMPLE

a(3)=10 because we have (U)D(U)D(U)D, (U)D(U)UDD, (U)UDD(U)D, (U)UDUDD and (U)U(U)DDD, the up steps starting at even level being shown between parentheses (U=(1,1), D=(1,1)).


MAPLE

G:=z*(13*z+z^2+5*z^35*z^4)/(1+z)/(13*z+z^2)^2/(1zz^2): Gser:=series(G, z=0, 34): seq(coeff(Gser, z, n), n=1..30);


MATHEMATICA

Rest[CoefficientList[Series[x*(13*x+x^2+5*x^35*x^4)/(1+x)/(13*x+x^2)^2 /(1xx^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



