A121484 -id:A121484 - cp's OEIS frontend

A121481 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 3, 6, 0, 1, 5, 14, 5, 9, 0, 1, 12, 22, 35, 7, 12, 0, 1, 22, 68, 53, 65, 9, 15, 0, 1, 49, 127, 203, 97, 104, 11, 18, 0, 1, 94, 329, 390, 444, 153, 152, 13, 21, 0, 1, 201, 664, 1157, 873, 816, 221, 209, 15, 24, 0, 1, 396, 1576, 2456, 2925, 1627, 1345

Offset: 0

Emeric Deutsch, Aug 02 2006

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A121482(n). Sum(k*T(n,k),k=0..n)=A121483(n).

			T(3,1)=3 because we have U|DUUDUDD, UUDUU|DDD and UUU|DDUDD, where U=(1,1) and D=(1,-1) (the peaks at odd level are shown by a |; the Dyck path UUDUDDUD has 1 peak at odd level but it is not nondecreasing).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
3,3,6,0,1;
5,14,5,9,0,1;

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, A121482, A121483, A121484.

G:=(t*z^2+z^2+z-1)*(t*z^3+2*z^2-1)/(1-4*z^2-z-t*z+2*z^4*t+4*z^4-z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) = (tz^2+z^2+z-1)(tz^3+2z^2-1)/(1-4z^2-z-tz+2tz^4+4z^4-z^6+2z^3+tz^3+t^2*z^3).

A121485 Number of nondecreasing Dyck paths of semilength n and having no peaks at even level (n>=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, 1, 2, 4, 8, 16, 33, 66, 136, 274, 562, 1138, 2327, 4725, 9645, 19613, 39997, 81397, 165906, 337773, 688260, 1401565, 2855432, 5815477, 11846941, 24129498, 49152840, 100116607, 203936639, 415394872, 846143795, 1723513075, 3510704795

Offset: 1

Emeric Deutsch, Aug 02 2006

Column 0 of A121484.

			a(4)=4 because we have UDUDUDUD, UDUUUDDD, UUUDDDUD and UUUDUDDD, where U=(1,1) and D=(1,-1).

Colin Barker, Table of n, a(n) for n = 1..1000
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 (1,4,-2,-4,0,1).

Cf. A121482, A121484.

I:=[1,1,2,4,8,16]; [n le 6 select I[n] else Self(n-1)+4*Self(n-2)-2*Self(n-3)-4*Self(n-4)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Sep 12 2015

G:=z*(1-z^2)*(1-2*z^2)/(1-4*z^2-z+4*z^4-z^6+2*z^3): Gser:=series(G,z=0,40): seq(coeff(Gser,z,n),n=1..37);

LinearRecurrence[{1, 4, -2, -4, 0, 1}, {1, 1, 2, 4, 8, 16}, 40] (* Vincenzo Librandi, Sep 12 2015 *)

Vec(z*(1-z^2)*(1-2*z^2)/(1-z-4*z^2+2*z^3+4*z^4-z^6) + O(z^60)) \\ Michel Marcus, Sep 11 2015

G.f.: z(1-z^2)(1-2z^2)/(1-z-4z^2+2z^3+4z^4-z^6).

a(n) = a(n-1)+4*a(n-2)-2*a(n-3)-4*a(n-4)+a(n-6) for n>6. - Colin Barker, Sep 11 2015

A121486 Number of peaks at 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.

Original entry on oeis.org

0, 1, 4, 13, 43, 132, 400, 1184, 3461, 9999, 28634, 81383, 229860, 645731, 1805582, 5028189, 13952221, 38590922, 106434540, 292792026, 803565215, 2200694791, 6015268164, 16412564173, 44708036568, 121600924117, 330277253560

Offset: 1

Emeric Deutsch, Aug 02 2006

nonn

			a(3)=4 because in UDUDUD, UDUU|DD, UU|DDUD, UU|DU|DD and UUUDDD we have altogether 4 peaks at even level (shown by a |); here U=(1,1) and 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.
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-5,15,-1,-4,1).

Cf. A121484, A121483, A038731.

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

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

a(n) = Sum(k*A121484(n,k),k=0..n-1).
G.f.: z^2*(1-z)(1-z-3z^2+3z^3-z^4)/[(1+z)(1-z-z^2)(1-3z+z^2)^2].
a(n) ~ (sqrt(5)-1) * (3+sqrt(5))^n * n / (5 * 2^(n+2)). - Vaclav Kotesovec, Mar 20 2014
20*a(n) = -8*(-1)^n +10*(2*A001871(n)-5*A001871(n-1))+5*(4*A000045(n+1)-7*A000045(n))-3*(4*A001906(n+1)+9*A001906(n)). - R. J. Mathar, Jul 26 2022

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A121481 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).

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A121485 Number of nondecreasing Dyck paths of semilength n and having no peaks at even level (n>=0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

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A121486 Number of peaks at 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.

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