A000957 -id:A000957 - cp's OEIS frontend

A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.

1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156

Offset: 0

Philippe Deléham, Mar 21 2007

sign

Hankel transform is (-1)^n.

Catalan transform of A033999. - R. J. Mathar, Nov 11 2008

Fung Lam, Table of n, a(n) for n = 0..1500
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Cf. A000108, A000957, A039599, A064310, A106566.

[1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]])): n in [1..30]]; // G. C. Greubel, Feb 27 2019

Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *)

{a(n) = (-1/2)^n*(1+sum(k=0,n-1, (-2)^k*binomial(2*k,k)/(k+1)))};
vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019

from itertools import count, islice
def A126983_gen(): # generator of terms
    yield from (1, -1, 0)
    a, c = 0, 1
    for n in count(1):
        yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1)
A126983_list = list(islice(A126983_gen(),20)) # Chai Wah Wu, Apr 27 2023

[1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # G. C. Greubel, Feb 27 2019

a(n) = (-1)^n*A064310(n).
a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k.
From Philippe Deléham, Nov 15 2009: (Start)
a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1.
a(n) = -A000957(n) for n>0. (End)
Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014
a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, 5583, 19683, 72162, 274796, 1082439, 4406706, 18484332, 79818616, 353995743, 1611041726, 7510754022, 35842380314, 174850257639, 871343536591, 4430997592209, 22978251206350, 121410382810005, 653225968918521

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 13 2014

nonn

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
Also the number of ballot sequences without two consecutive elements equal. A ballot sequence B is a string such that, for all prefixes P of B, h(i)>=h(j) for iA000085).
First column (k=0) of A238125.

			The a(5) = 9 such tableaux of 5 are:
[1]   [2]  [3]   [4]  [5]  [6]  [7]  [8]  [9]
135   13   135   13   13   14   14   15   1
24    24   2     25   2    25   2    2    2
      5    4     4    4    3    3    3    3
                      5         5    4    4
                                          5
The corresponding ballot sequences are:
1:  [ 0 1 0 1 0 ]
2:  [ 0 1 0 1 2 ]
3:  [ 0 1 0 2 0 ]
4:  [ 0 1 0 2 1 ]
5:  [ 0 1 0 2 3 ]
6:  [ 0 1 2 0 1 ]
7:  [ 0 1 2 0 3 ]
8:  [ 0 1 2 3 0 ]
9:  [ 0 1 2 3 4 ]

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..68 (terms 0..48 from Alois P. Heinz)
Timothy Y. Chow, Henrik Eriksson and C. Kenneth Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol.11, no.2, (2005).
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Cf. A000085 (all Young tableaux), A000957, A001181, A214021, A214087, A214159, A214875.
Cf. A238126 (tableaux with one succession), A238127 (two successions).
Cf. A264051, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [i, l[]]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..30);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember;
      `if`(n<1, 1, add(`if`(i<>v and (i=1 or l[i-1]>l[i]),
       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
       b(n-1, nops(l)+1, [l[], 1]))
    end:
a:= proc(n) option remember; forget(b); b(n-1, 1, [1]) end:
seq(a(n), n=0..30);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i != v && (i == 1 || l[[i-1]] > l[[i]]), b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; a[n_] := a[n] = b[n-1, 1, {1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, translated from 2nd Maple program *)

a(n) = Sum_{k=1..A264078(n)} k * A264051(n,k). - Alois P. Heinz, Nov 02 2015

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).

Original entry on oeis.org

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

A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 6, 6, 1, 1, 1, 0, 1, 22, 72, 18, 1, 1, 1, 0, 1, 92, 1289, 960, 57, 1, 1, 1, 0, 1, 422, 29889, 93964, 14257, 186, 1, 1, 1, 0, 1, 2074, 831174, 13652068, 8203915, 228738, 622, 1, 1

Offset: 0

Alois P. Heinz, Jul 01 2012

A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.

			A(2,4) = 1:
  [1 3 5 7]
  [2 4 6 8].
A(4,2) = 6:
  [1, 5]   [1, 4]   [1, 3]   [1, 4]   [1, 3]   [1, 3]
  [2, 6]   [2, 6]   [2, 6]   [2, 5]   [2, 5]   [2, 4]
  [3, 7]   [3, 7]   [4, 7]   [3, 7]   [4, 7]   [5, 7]
  [4, 8]   [5, 8]   [5, 8]   [6, 8]   [6, 8]   [6, 8].
Square array A(n,k) begins:
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  0,     0,       0,          0,              0, ...
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  2,     6,      22,         92,            422, ...
  1, 1,  6,    72,    1289,      29889,         831174, ...
  1, 1, 18,   960,   93964,   13652068,     2621897048, ...
  1, 1, 57, 14257, 8203915, 8134044455, 11865331748843, ...

Alois P. Heinz, Antidiagonals n = 0..23, flattened
T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Rows n=0+2, 3-4 give: A000012, A001181(k) for k>0, A214875.
Columns k=0+1, 2, 3 give: A000012, A000957(n+1), A214159.
Main diagonal gives A264103.
Cf. A214020, A214088.

b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
       add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
    end:
A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, l}]}; If[s == 0, 1, Sum[If[t != i && l[[i]] > If[i == n, 0, l[[i+1]]], b[ReplacePart[l, i -> l[[i]]-1], i], 0], {i, 1, n}]] ] ; a[n_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A065601 Number of Dyck paths of length 2n with exactly 1 hill.

Original entry on oeis.org

0, 1, 0, 2, 4, 13, 40, 130, 432, 1466, 5056, 17672, 62460, 222853, 801592, 2903626, 10582816, 38781310, 142805056, 528134764, 1960825672, 7305767602, 27307800400, 102371942932, 384806950624, 1450038737668, 5476570993440, 20727983587220, 78606637060012

Offset: 0

N. J. A. Sloane, Dec 02 2001

Convolution of A000957(n) with itself gives a(n-1).

Alois P. Heinz, Table of n, a(n) for n = 0..500
Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)

2nd column of A065600. Cf. A000957.

b:= proc(x, y, h, z) option remember;
     `if`(x<0 or y b(n$2, true$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2012
# second Maple program:
series(((1-sqrt(1-4*x))/(3-sqrt(1-4*x)))^2/x, x=0, 30);  # Mark van Hoeij, Apr 18 2013

CoefficientList[Series[((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))^2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
Table[Sum[(-1)^j*(j+1)*(j+2)*Binomial[2*n-1-j,n],{j,0,n-1}]/(n+1),{n,0,30}] (* Vaclav Kotesovec, May 18 2015 *)

Reference gives g.f.'s.
Conjecture: 2*(n+1)*a(n) +(-3*n+2)*a(n-1) +2*(-9*n+19)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Dec 10 2013
a(n) ~ 2^(2*n+3) / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

More terms from Emeric Deutsch, Dec 03 2001

A104629 Expansion of (1-2x-sqrt(1-4x))/(x^2 * (1+2x+sqrt(1-4x))).

Original entry on oeis.org

1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, 91144, 325878, 1174281, 4260282, 15548694, 57048048, 210295326, 778483932, 2892818244, 10786724388, 40347919626, 151355847012, 569274150156, 2146336125648, 8110508473252

Offset: 0

Paul Barry, Mar 17 2005

Diagonal sums of A039598.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A000957, A039598, A064310.

Partial sums of A122920.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt(1-4*x))/(x^2*(1+2*x+Sqrt(1-4*x))))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[((1-2x-Sqrt[1-4x])/(1+2x+Sqrt[1-4x]))/x^2,{x,0,30}],x] (* Harvey P. Dale, Jul 23 2016 *)
Table[(1 + Sum[CatalanNumber[n]*(-2)^k, {k,0,n+2}])/(8*(-2)^n), {n,0,30}] (* G. C. Greubel, Aug 12 2018 *)

x='x+O('x^30); Vec((1-2*x-sqrt(1-4*x))/(x^2*(1+2*x+sqrt(1-4*x)))) \\ G. C. Greubel, Aug 12 2018

for(n=0,30, print1((1 + sum(k=0,n+2, (-2)^k*binomial(2*k, k)/(k+1)))/(8*(-2)^n), ", ")) \\ G. C. Greubel, Aug 12 2018

from itertools import count, islice
def A104629_gen(): # generator of terms
    a, c = 0, 1
    for n in count(1):
        yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)
A104629_list = list(islice(A104629_gen(),20)) # Chai Wah Wu, Apr 26 2023

a(n) = A000957(n+3).
a(n) = (1 + Sum_{k=0..n+2} C(k)*(-2)^k)/(8*(-2)^n), where C(n) = Catalan numbers.
D-finite with recurrence: 2*(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(-2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 30 2014 [Verified by Georg Fischer, Apr 27 2023]

A158815 Triangle T(n,k) read by rows, matrix product of A046899(row-reversed) * A130595.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 46, 16, 6, 1, 1, 166, 58, 19, 7, 1, 1, 610, 211, 71, 22, 8, 1, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1, 32206, 11006, 3676, 1196, 376, 116, 31, 11, 1, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 27 2009

The left factor of the matrix product is the triangle which starts
   1;
   2,  1;
   6,  3, 1;
  20, 10, 4, 1;
a row-reversed version of A046899, equivalent to the triangular view of the array A092392. The right factor is the inverse of the matrix A007318, which is A130595.
Swapping the two factors, A007318^(-1) * A046899(row-reversed) would generate A158793.
Riordan array (f(x), g(x)) where f(x) is the g.f. of A026641 and where g(x) is the g.f. of A000957. - Philippe Deléham, Dec 05 2009
T(n,k) is the number of nonnegative paths consisting of upsteps U=(1,1) and downsteps D=(1,-1) of length 2n with k low peaks. (A low peak has its peak vertex at height 1.) Example: T(3,1)=5 counts UDUUUU, UDUUUD, UDUUDU, UDUUDD, UUDDUD. - David Callan, Nov 21 2011
Matrix product P^2 * Q * P^(-2), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A171243. - Peter Bala, Jul 13 2021

			The triangle starts
       1;
       1,     1;
       4,     1,     1;
      13,     5,     1,    1;
      46,    16,     6,    1,    1;
     166,    58,    19,    7,    1,   1;
     610,   211,    71,   22,    8,   1,   1;
    2269,   781,   261,   85,   25,   9,   1,  1;
    8518,  2620,   976,  316,  100,  28,  10,  1,  1;
   32206, 11006,  3676, 1196,  376, 116,  31, 11,  1, 1;
  122464, 41746, 13938, 4544, 1442, 441, 133, 34, 12, 1, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Filippo Disanto, Andrea Frosini and Simone Rinaldi, Renzo Pinzani, The Combinatorics of Convex Permutominoes, Southeast Asian Bulletin of Mathematics (2008) 32: 883-912.

Cf. A000984, A026641, A046899, A158793, A171243.

A158815 := proc (n, k)
  add((-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k), j = 0..n);
end proc:
seq(seq(A158815(n, k), k = 0..n), n = 0..10); # Peter Bala, Jul 13 2021

T[n_,k_]:= T[n,k]= Sum[(-1)^(j+k)*Binomial[j,k]*Binomial[2*n-j,n], {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)

def A158815(n,k): return sum( (-1)^(j+k)*binomial(2*n-j, n)*binomial(j, k) for j in (0..n) )
flatten([[A158815(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

Sum_{k=0..n} T(n,k) = A046899(n).
T(n,0) = A026641(n).
Sum_{k=0..n} T(n,k)*x^k = A026641(n), A000984(n), A001700(n), A000302(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Dec 03 2009
T(n, k) = Sum_{j=0..n} binomial(j, k)*binomial(2*n-j, n). - Peter Bala, Jul 13 2021

A059019 Number of Dyck paths of semilength n with no peak at height 3.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 58, 163, 483, 1494, 4783, 15740, 52956, 181391, 630533, 2218761, 7888266, 28291588, 102237141, 371884771, 1360527143, 5002837936, 18479695171, 68539149518, 255137783916, 952914971191, 3569834343547, 13410481705795

Offset: 0

N. J. A. Sloane, Feb 12 2001

nonn

From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is the upper left term in M^n, where M is an infinite square production matrix in which a column of (1,1,0,0,0,...) is prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  0, 1, 1, 1, 0, 0, ...
  0, 1, 1, 1, 1, 0, ...
  0, 1, 1, 1, 1, 1, ...
  ... (End)

			1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 58*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.

G_1, G_2, G_3, G_4 give A000957, A000108, A059019, A059027, respectively.

A059019:=n->add(add(binomial(k,j)*j*binomial(2*n-2*k-j-1,n-k-j)/(n-k),j=0..min(k,n-k)), k=1..n-1)+1: seq(A059019(n),n=0..30); # Wesley Ivan Hurt, Oct 01 2014

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

a(n):=sum(sum(binomial(k,j)*j*binomial(2*n-2*k-j-1,n-k-j),j,0,min(k,n-k))/(n-k),k,1,n-1)+1; \\ Vladimir Kruchinin, Oct 02 2013

x='x+O('x^66); Vec( 2/(2-3*x+x*(1-4*x)^(1/2)) ) \\ Joerg Arndt, Oct 02 2013

G.f.: 2/(2 - 3*x + x*(1-4*x)^(1/2)).
a(n) = Sum_{k=1..n-1} (Sum_{j=0..min(k,n-k)} binomial(k,j)*j*binomial(2*n-2*k-j-1, n-k-j) /(n-k)) + 1. - Vladimir Kruchinin, Oct 02 2013
a(n) ~ 2^(2*n + 2)/(25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013
a(n+1) - a(n) = A135337(n), n > 0. - Philippe Deléham, Oct 02 2014

A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 6, 3, 1, 24, 18, 10, 4, 1, 75, 57, 33, 15, 5, 1, 243, 186, 111, 54, 21, 6, 1, 808, 622, 379, 193, 82, 28, 7, 1, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1, 33062, 25724, 16266, 8928, 4332, 1856, 692, 218, 55, 10, 1

Offset: 2

N. J. A. Sloane, Dec 02 2001

A Riordan triangle.
Subtriangle of triangle in A167772. - Philippe Deléham, Nov 14 2009
Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A000958 and g(x) is the g.f. of A000108. - Philippe Deléham, Jan 23 2010

			T(3,2)=1 reflecting the unique Dyck path (UUDUDD) of length 6, with no hills and height of first peak equal to 2.
Triangle begins:
     1;
     1,    1;
     3,    2,    1;
     8,    6,    3,   1;
    24,   18,   10,   4,   1;
    75,   57,   33,  15,   5,   1;
   243,  186,  111,  54,  21,   6,  1;
   808,  622,  379, 193,  82,  28,  7,  1;
  2742, 2120, 1312, 690, 311, 118, 36,  8,  1;

Reinhard Zumkeller, Rows n = 2..125 of triangle, flattened
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Row sums give A000957 (the Fine sequence).
First column is A000958.
Cf. A000108, A007318, A167772.

a065602 n k = sum
   [(k-1+2*j) * a007318' (2*n-k-1-2*j) (n-1) `div` (2*n-k-1-2*j) |
    j <- [0 .. div (n-k) 2]]
a065602_row n = map (a065602 n) [2..n]
a065602_tabl = map a065602_row [2..]
-- Reinhard Zumkeller, May 15 2014

a := proc(n,k) if n=0 and k=0 then 1 elif k<2 or k>n then 0 else sum((k-1+2*j)*binomial(2*n-k-1-2*j,n-1)/(2*n-k-1-2*j),j=0..floor((n-k)/2)) fi end: seq(seq(a(n,k),k=2..n),n=1..14);

nmax = 12; t[n_, k_] := Sum[(k-1+2j)*Binomial[2n-k-1-2j, n-1] / (2n-k-1-2j), {j, 0, (n-k)/2}]; Flatten[ Table[t[n, k], {n, 2, nmax}, {k, 2, n}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)

def T(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
flatten([[T(n,k) for k in (2..n)] for n in (2..12)]) # G. C. Greubel, May 26 2022

T(n, 2) = A000958(n-1).
Sum_{k=2..n} T(n, k) = A000957(n+1).
From Emeric Deutsch, Feb 23 2004: (Start)
T(n, k) = Sum_{j=0..floor((n-k)/2)} (k-1+2*j)*binomial(2*n-k-1-2*j, n-1)/(2*n-k-1-2*j).
G.f.: t^2*z^2*C/( (1-z^2*C^2)*(1-t*z*C) ), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
T(n,k) = A167772(n-1,k-1), k=2..n. - Reinhard Zumkeller, May 15 2014
From G. C. Greubel, May 26 2022: (Start)
T(n, n-1) = A000027(n-2).
T(n, n-2) = A000217(n-2).
T(n, n-3) = A166830(n-3). (End)

More terms from Emeric Deutsch, Feb 23 2004

A114587 Number of peaks at odd levels in all hill-free Dyck paths of semilength n+3 (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 4, 17, 68, 269, 1056, 4132, 16144, 63046, 246228, 962019, 3760700, 14710589, 57581696, 225546488, 884059808, 3467476430, 13608852968, 53443415522, 210000136136, 825630208466, 3247733377664, 12781815016232, 50328168273408

Offset: 0

Emeric Deutsch, Dec 11 2005

nonn

			a(1)=4 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UU(UD)(UD)DD, UUDU(UD)DD, UUDUDUDD, UU(UD)DUDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 4 peaks at odd level (shown between parentheses).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A114586, A114590, A114515.

G:=(1-2*z-3*z^2-2*z^3-(1-z^2)*sqrt(1-4*z))/2/sqrt(1-4*z)/z^4/(2+z)^2: Gser:=series(G,z=0,32): 1,seq(coeff(Gser,z^n),n=1..26);

CoefficientList[Series[(1-2*x-3*x^2-2*x^3-(1-x^2)*Sqrt[1-4*x])/(2*x^4*(2+x)^2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

G.f.: (1 - 2*x - 3*x^2 - 2*x^3 - (1 - x^2)*sqrt(1 - 4*x))/(2*x^4*(2 + x)^2 * sqrt(1 - 4*x)).
a(n) = Sum_{k=0..n+1} k*A114586(n+3,k).
Recurrence: 8*n*(n+4)*a(n) = 2*(15*n^2 + 47*n + 18)*a(n-1) + (9*n^2 + 70*n + 80)*a(n-2) - 2*(n+1)*(2*n+1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+6)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

cp's OEIS Frontend

A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108.

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A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

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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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A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A065601 Number of Dyck paths of length 2n with exactly 1 hill.

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A104629 Expansion of (1-2*x-sqrt(1-4*x))/(x^2 * (1+2*x+sqrt(1-4*x))).

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A104629 Expansion of (1-2x-sqrt(1-4x))/(x^2 * (1+2x+sqrt(1-4x))).