A121529 -id:A121529 - cp's OEIS frontend

A121530 Number of double rises at an odd 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.

0, 1, 4, 14, 47, 148, 454, 1359, 4004, 11644, 33521, 95696, 271300, 764605, 2143964, 5985186, 16643779, 46124692, 127433562, 351106955, 964976460, 2646158176, 7241414949, 19779499584, 53933402472, 146828245753, 399137621524

Offset: 1

Emeric Deutsch, Aug 05 2006

nonn

a(n)=Sum(k*A121529(n,k), k>=0). a(n)+A121532(n)=A054444(n-2).

			a(3)=4 because we have UDUDUD, UDU/UDD, U/UDDUD, U/UDUDD and U/UUDDD, the double rises at an odd level being indicated by a / (U=(1,1), D=(1,-1)).

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1)

Cf. A121529, A121532, A054444.

g:=z^2*(1-2*z-z^2+4*z^3-3*z^4)/(1+z)/(1-3*z+z^2)^2/(1-z-z^2): gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=1..30);

Rest[CoefficientList[Series[x^2*(1-2*x-x^2+4*x^3-3*x^4)/(1+x)/(1-3*x+x^2)^2 /(1-x-x^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 20 2014 *)

G.f.=z^2*(1-2z-z^2+4z^3-3z^4)/[(1+z)(1-3z+z^2)^2*(1-z-z^2)].
a(n) ~ (3-sqrt(5)) * (3+sqrt(5))^n * n / (5 * 2^(n+1)). - Vaclav Kotesovec, Mar 20 2014
Equivalently, a(n) ~ phi^(2*n-2) * n / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

A121531 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an even level (n >= 1, k >= 0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

Original entry on oeis.org

1, 2, 4, 1, 7, 6, 12, 20, 2, 20, 51, 18, 33, 115, 80, 5, 54, 240, 262, 54, 88, 477, 725, 294, 13, 143, 916, 1803, 1158, 161, 232, 1716, 4170, 3768, 1026, 34, 376, 3155, 9152, 10815, 4684, 475, 609, 5717, 19311, 28418, 17432, 3449, 89, 986, 10240, 39520

Offset: 1

Emeric Deutsch, Aug 05 2006

Row n contains ceiling(n/2) terms.
Row sums are the odd-indexed Fibonacci numbers (A001519).
T(n,0) = Fibonacci(n+2) - 1 = A000071(n+2).
Sum_{k>=0} k*T(n,k) = A121532(n).

			T(5,2)=2 because we have UU/UU/UDDDDD and UU/UDDU/UDDD, where U=(1,1) and D=(1,-1) (the double rises at an even level are indicated by a /).
Triangle starts:
   1;
   2;
   4,   1;
   7,   6;
  12,  20,  2;
  20,  51, 18;
  33, 115, 80, 5;

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

Cf. A001519, A121529, A121532.

G:=z*(1-2*t*z^2-t*z^3)*(1-t*z^2)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=0..ceil(n/2)-1) od; # yields sequence in triangular form

G.f.: G = G(t,z) = z(1 - 2tz^2 - tz^3)(1-tz^2)/((1 - z - tz^2)(1 - z - z^2 - 3tz^2 - tz^3 + t^2*z^4)).

cp's OEIS Frontend

A121530 Number of double rises at an odd 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.

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