A118972 -id:A118972 - cp's OEIS frontend

A001558 Number of hill-free Dyck paths of semilength n+3 and having length of first descent equal to 1 (a hill in a Dyck path is a peak at level 1).

1, 3, 10, 33, 111, 379, 1312, 4596, 16266, 58082, 209010, 757259, 2760123, 10114131, 37239072, 137698584, 511140558, 1904038986, 7115422212, 26668376994, 100221202998, 377570383518, 1425706128480, 5394898197448, 20454676622476

Offset: 0

N. J. A. Sloane

a(n) is also the number of even-length descents to ground level in all Dyck paths of semilength n+2. Example: a(1)=3 because in UDUDUD, UDUU(DD), UU(DD)UD, UUDU(DD) and UUUDDD we have 3 even-length descents to ground level (shown between parentheses). - Emeric Deutsch, Oct 05 2008
Convolution of A000108 with A104629. - Philippe Deléham, Nov 11 2009
The Kn12 triangle sums of A039599 are given by the terms of this sequence. For the definition of this and other triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

			a(1)=3 because we have uu(d)ududd, uuu(d)uddd and uu(d)uuddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
G.f. = 1 + 3*x + 10*x^2 + 33*x^3 + 111*x^4 + 379*x^5 + 1312*x^6 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
T. Fine, Extrapolation when very little is known about the source, Information and Control 16 (1970), 331-359.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

Cf. A000957, A118972, A118973.

Cf. A111301. - Emeric Deutsch, Oct 05 2008

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(x*(3-Sqrt(1-4*x)))*((1-Sqrt(1-4*x))/(2*x))^3 )); // G. C. Greubel, Feb 12 2019

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=F*C^3: gser:=series(g,z=0,32): seq(coeff(gser,z,n),n=0..27); # Emeric Deutsch, May 08 2006

CoefficientList[Series[(1-Sqrt[1-4*x])/(x*(3-Sqrt[1-4*x]))*((1-Sqrt[1-4*x])/(2*x))^3, {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3) \\ G. C. Greubel, Feb 12 2019

((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

a(n) = A000957(n+4) - A000957(n+3) - A000957(n+2) (A000957 are the Fine numbers). - Emeric Deutsch, May 08 2006
a(n) = A118972(n+3,1). - Emeric Deutsch, May 08 2006
G.f.: F*C^3, where F = (1-sqrt(1-4z))/(z*(3-sqrt(1-4z))) and C = (1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, May 08 2006
a(n) = Sum_{k>=0} k*A111301(n+2,k). - Emeric Deutsch, Oct 05 2008
(n+3)*a(n) = (-(11/2)*n + 21/2)*a(n-3) + ((9/2)*n + 11/2)*a(n-1) + (-(1/2)*n + 9/2)*a(n-2) + (-2n + 5)*a(n-4). - Simon Plouffe, Feb 09 2012
a(n) ~ 11*2^(2*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Edited by Emeric Deutsch, May 08 2006

A118973 Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 0, 2, 5, 16, 51, 168, 565, 1934, 6716, 23604, 83806, 300154, 1083137, 3934404, 14374413, 52787766, 194746632, 721435884, 2682522918, 10008240456, 37455101382, 140569122624, 528926230530, 1994980278636, 7541234323096

Offset: 0

Emeric Deutsch, May 08 2006

nonn

Also, for a given j>=2, number of hill-free Dyck paths of semilength n+j and having length of first descent equal to j. a(n)=A000108(n+1)-A000108(n)-[A000957(n+2)-A000957(n+1)]. Columns 2,3,4,... of A118972 (without the initial 0's).

			a(2)=2 because we have uu(dd)uudd and uuu(dd)udd, where u=(1,1),d=(1,-1) (the first descents are shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A000108, A000957, A118972, A001558.

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=(1-z)*C*F: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..28);
A118973List := proc(m) local A, P, n; A := [1,0]; P := [1,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
A := [op(A), P[-1]] od; A end: A118973List(26); # Peter Luschny, Mar 26 2022

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

a(n):=(sum((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1)),k,0,n-1))+(-1)^(n); /* Vladimir Kruchinin, Mar 06 2016 */

my(x='x+O('x^25)); Vec((1-x)*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1-sqrt(1-4*x))/2/x) \\ G. C. Greubel, Feb 08 2017

G.f.: (1-x)*C*F, where F = (1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x))) and C = (1-sqrt(1-4*x))/(2*x) is the Catalan function.
a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
a(n) = (Sum_{k=0..n-1}((k+2)*(-1)^k*(binomial(2*n-k+1,n-k)/(n+2)-binomial(2*n-k-1,n-k-1)/(n+1))))+(-1)^(n). - Vladimir Kruchinin. Mar 06 2016
D-finite with recurrence +2*(n+2)*a(n) +(-7*n-2)*a(n-1) +2*(-3*n+1)*a(n-2) +(7*n-26)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A118974 Sum of the lengths of the first descents in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

0, 0, 2, 4, 11, 31, 94, 298, 977, 3283, 11243, 39087, 137569, 489171, 1754596, 6340756, 23063731, 84372061, 310216081, 1145748061, 4248861631, 15814069951, 59054807821, 221197379221, 830819449003, 3128511421663, 11808294045071, 44666151392095, 169294875129839

Offset: 0

Emeric Deutsch, May 08 2006

nonn

			a(4)=11 because in the hill-free Dyck paths of semilength 4, namely uu(dd)uudd, uu(d)uuddd, uu(d)ududd, uuu(dd)udd, uuu(d)uddd and uuuu(dddd), the sum of the lengths of the first descents (shown between parentheses) is 2+1+1+2+1+4=11.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A000957, A118972.

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=series(z^2*C*F*(1+C-z*C)/(1-z),z=0,32): seq(coeff(g,z,n),n=0..28);

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

my(x='x+O('x^50)); concat([0,0], Vec(x^2*(1-sqrt(1-4*x))/2/x*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1+(1-sqrt(1-4*x))/2/x-x*(1-sqrt(1-4*x))/2/x)/(1-x))) \\ G. C. Greubel, Mar 18 2017

a(n) = Sum_{k=1,..,n} k*A118972(n,k).
G.f.: z^2*C*F*(1+C-z*C)/(1-z), where F = (1-sqrt(1-4*z))/(z*(3-sqrt(1-4*z))) and C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) ~ 17*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: 2*(n+1)*(17*n^2-65*n+60)*a(n) -3*(3*n-4)*(17*n^2-48*n+15)*a(n-1) +3*(17*n^3-82*n^2+121*n-60)*a(n-2) +2*(2*n-5) *(17*n^2-31*n+12) *a(n-3)=0. - R. J. Mathar, Jun 22 2016

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A001558 Number of hill-free Dyck paths of semilength n+3 and having length of first descent equal to 1 (a hill in a Dyck path is a peak at level 1).

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A118973 Number of hill-free Dyck paths of semilength n+2 and having length of first descent equal to 2 (a hill in a Dyck path is a peak at level 1).

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A118974 Sum of the lengths of the first descents in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

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