A000698 -id:A000698 - cp's OEIS frontend

A258174 Sum over all Dyck paths of semilength n of products over all peaks p of x_p*y_p, where x_p and y_p are the coordinates of peak p.

1, 1, 7, 84, 1486, 35753, 1111931, 43150593, 2035666985, 114412223081, 7538224510181, 574552299138202, 50096579094908148, 4949493445607316419, 549534510282406667069, 68071071679372210762156, 9347203754680124767253730, 1414740620049957735248175695

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258175, A258176, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x*y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x*y, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 2, 12, 114, 1448, 22770, 424164, 9095450, 220023184, 5914998594, 174682531260, 5614908340866, 194967208057272, 7267467723747218, 289270983756577620, 12239218862861690250, 548301077168477951520, 25918121712918957399426, 1288797080051656060595820

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..350
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258176, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x+y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x + y, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258176 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^y_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 7, 142, 9354, 2503597, 3260627607, 24105227716863, 1028836978599566213, 290383808553140390346475, 511963364817949502725911280781, 6704846980724405836568589845161191576, 584709361918378923208855262622537662297053728

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..45
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258177, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^y, 1] + b[x - 1, y + 1, True]]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 5, 112, 15312, 22928885, 475971133797, 164769697242392241, 1674694178196441599627207, 434453335415659344048321288040053, 2772047111897899211702422870954450438220795, 919691726760748842849028933552012720445531166591469510

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..35
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258178, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, y^x, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, y^x, 1] + b[x - 1, y + 1, True]]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.

Original entry on oeis.org

1, 1, 13, 414, 24324, 2279209, 311524201, 58467947511, 14424374692879, 4525566110365523, 1759527523008436279, 830255082140922306224, 467382831980334193769718, 309419146352957449765072455, 237980526477430552734199922151, 210427994109788912088395561755374

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258179, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x^2, 1] + b[x - 1, y + 1, True] ]];
a[n_] :=  b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

Original entry on oeis.org

1, 1, 5, 34, 312, 3649, 52161, 889843, 17796555, 411120395, 10838039407, 322752018060, 10762432731362, 398802951148255, 16312276452291935, 732189190349581890, 35876807697443520000, 1910107567584518883891, 110035833179472385285367, 6832792252684597270659486

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258180, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, y^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/( (k+2)^2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

G.f.: T(0), where T(k) = 1 - x/( (k+2)^2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015

A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 4, 33, 517, 15326, 852912, 91023697, 19716262702, 8794395041567, 8016790849841585, 15556074485786226848, 64891787190080888991273, 561815453349204340865790817, 10402242033224422585780623039909, 423787530114579490372987256671625678

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..75
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258181.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, binomial(x, y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, Binomial[x, y], 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 5, 89, 5933, 1540161, 1584150165, 6497470064169, 106497075348688637, 6980195689972655145233, 1829876050804408046228327525, 1918781572083632396857805205324025, 8047973452254281276702044410544321359565, 135022681866797995009325363468217320506328688097

Offset: 0

Alois P. Heinz, May 22 2015

nonn

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..55
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, 2^(x-y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, 2^(x - y), 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

a(n) ~ c * 2^(n*(n-1)), where c = 1.47818066525747143617276638534... . - Vaclav Kotesovec, Jun 01 2015

A053979 Triangle T(n,k) giving number of rooted maps regardless of genus with n edges and k nodes (n >= 0, k = 1..n+1).

Original entry on oeis.org

1, 1, 1, 3, 5, 2, 15, 32, 22, 5, 105, 260, 234, 93, 14, 945, 2589, 2750, 1450, 386, 42, 10395, 30669, 36500, 22950, 8178, 1586, 132, 135135, 422232, 546476, 388136, 166110, 43400, 6476, 429, 2027025, 6633360, 9163236, 7123780, 3463634, 1092560, 220708, 26333, 1430

Offset: 0

N. J. A. Sloane, Apr 09 2000

Triangle T(n,k), read by rows, given by (1,2,3,4,5,6,7,8,9,...) DELTA (1,1,1,1,1,1,1,1,1,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 21 2011.

A127160*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 2012

			A(x;t) = t + (t + t^2)*x + (3*t + 5*t^2 + 2*t^3)*x^2 + (15*t + 32*t^2 + 22*t^3 + 5*t^4)*x^3 + ...
Triangle begins :
n\k [1]     [2]     [3]     [4]     [5]     [6]    [7]   [8]
[0] 1;
[1] 1,      1;
[2] 3,      5,      2;
[3] 15,     32,     22,     5;
[4] 105,    260,    234,    93,     14;
[5] 945,    2589,   2750,   1450,   386,    42;
[6] 10395,  30669,  36500,  22950,  8178,   1586,  132;
[7] 135135, 422232, 546476, 388136, 166110, 43400, 6476, 429;
[8] ...

Gheorghe Coserea, Rows n=0..100, flattened
D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12.
R. Cori, Indecomposable permutations, hypermaps and labeled Dyck paths, arXiv:0812.0440v1 [math.CO], 2008.
R. Cori, Indecomposable permutations, hypermaps and labeled Dyck paths, Journal of Combinatorial Theory, Series A 116 (2009) 1326-1343.
J. Courtiel, K. Yeats, Connected chord diagrams and bridgeless maps, arXiv:1611.04611, eq. (18)
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218, eq. (5).

Cf. A000108, A000346, A001147, A084938. A000698, A025192, A084120, A127160, A167872.

G:=t/(1-(t+1)*z/(1-(t+2)*z/(1-(t+3)*z/(1-(t+4)*z/(1-(t+5)*z/(1-(t+6)*z/(1-(t+7)*z/(1-(t+8)*z/(1-(t+9)*z/(1-(t+10)*z/(1-(t+11)*z/(1-(t+12)*z)))))))))))):Gser:=simplify(series(G,z=0,10)):P[0]:=t:for n from 1 to 9 do P[n]:=sort(expand(coeff(Gser,z^n))) od:seq(seq(coeff(P[n],t^k),k=1..n+1),n=0..9); # Emeric Deutsch, Apr 01 2005

g = t/Fold[1-((t+#2)*z)/#1&, 1, Range[12, 1, -1]]; T[n_, k_] := SeriesCoefficient[g, {z, 0, n}, {t, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

A053979_ser(N,t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++, y1 = (1 + t*x*y0^2 + 2*x^2*y0')/(1-x);
    if (y1 == y0, break()); y0 = y1); y0;
};
concat(apply(p->Vecrev(p), Vec(A053979_ser(10))))
\\ test: y=A053979_ser(50); 2*x^2*deriv(y,x) == -t*x*y^2 + (1-x)*y - 1
\\ Gheorghe Coserea, May 31 2017

A053979_seq(N) = {
  my(t='t, R=vector(N), S=vector(N)); R[1]=S[1]=t;
  for (n=2, N,
    R[n] = t*subst(S[n-1],t,t+1);
    S[n] = R[n] + sum(k=1, n-1, R[k]*S[n-k]));
  apply(p->Vecrev(p), R/t);
};
concat(A053979_seq(10))
\\ test: y=t*Ser(apply(p->Polrev(p,'t), A053979_seq(50)),'x); y == t + x*y^2 + x*y + 2*x^2*deriv(y,x) && y == t + x*y*subst(y,t,t+1) \\ Riccati eq && Dyck eq
\\ Gheorghe Coserea, May 31 2017

G.f.: t/(1-(t+1)z/(1-(t+2)z/(1-(t+3)z/(1-(t+4)z/(1-(t+5)z/(1-... (Eq. (5) in the Arques-Beraud reference). - Emeric Deutsch, Apr 01 2005
Sum_{k = 0..n} (-1)^k*2^(n-k)*T(n,k) = A128709(n). Sum_{k = 0..n} T(n,k) = A000698(n+1). - Philippe Deléham, Mar 24 2007
From Peter Bala, Dec 22 2011: (Start)
The o.g.f. in the form G(x,t) = x/(1 - (t+1)*x^2/(1 - (t+2)*x^2/(1 - (t+3)*x^2/(1 - (t+4)*x^2/(1 - ... ))))) = x + (1+t)*x^3 + (3+5*t+2*t^2)*x^5 + ... satisfies the Riccati equation (1 - t*x*G)*G = x + x^3*dG/dx. The cases t = 0, t = 1 and t = 2 give A001147, A000698 and A167872, respectively. The cases t = -2, t = -3 and t = -4 give rational generating functions for aerated and signed versions of A000012, A025192 and A084120, respectively.
The identity G(x,1+t) = 1/(1+t)(1/x-1/G(x,t)) provided t <> -1 allows us to express G(x,n), n = 1,2,..., in terms of G(x,0), a generating function for the double factorial numbers.
Writing G(x,t) = Sum_{n >= 1} R(n,t)*x^(2*n-1), the row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) = (2*n-1)*R(n,t) + t*sum {k = 1..n} R(k,t)*R(n+1-k,t) with initial condition R(1,t) = 1.
G(x,t-1) = x + t*x^3 + (t+2*t^2)*x^5 + (3*t+7*t^2+5*t^3)*x^7 + ... is an o.g.f. for A127160.
The function b(x,t) = - t*G(1/x,t) satisfies the partial differential equation d/dx(b(x,t)) = -(t + (x + b(x,t))*b(x,t)). Hence the differential operator (D^2 + x*D + t), where D = d/dx, factorizes as (D - a(x,t))*(D - b(x,t)), where a(x,t) = -(x + b(x,t)). In the particular case t = -n, a negative integer, the functions a(x,-n) and b(x,-n) become rational functions of x expressible as the ratio of Hermite polynomials.
(End)

More terms from Emeric Deutsch, Apr 01 2005

A105623 Matrix square-root of triangle A105615.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 26, 10, 3, 1, 226, 74, 19, 4, 1, 2426, 706, 167, 31, 5, 1, 30826, 8162, 1831, 320, 46, 6, 1, 451586, 110410, 23843, 4021, 548, 64, 7, 1, 7489426, 1708394, 358339, 59024, 7801, 866, 85, 8, 1, 138722426, 29752066, 6097607, 987763, 127985

Offset: 0

Paul D. Hanna, Apr 16 2005

Column 0 equals A105616 (=column 1 of A105615) shift 1 place right. Column 1 is A000698 (related to double factorials) offset 1.

			Triangle begins:
1;
1,1;
4,2,1;
26,10,3,1;
226,74,19,4,1;
2426,706,167,31,5,1;
30826,8162,1831,320,46,6,1;
451586,110410,23843,4021,548,64,7,1;
7489426,1708394,358339,59024,7801,866,85,8,1;
138722426,29752066,6097607,987763,127985,13801,1289,109,9,1; ...

Cf. A105615, A105616 (column 0), A000698 (column 1), A105620 (matrix inverse).

T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))^-1); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

cp's OEIS Frontend

A258174 Sum over all Dyck paths of semilength n of products over all peaks p of x_p*y_p, where x_p and y_p are the coordinates of peak p.

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A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

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A258176 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^y_p, where x_p and y_p are the coordinates of peak p.

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A258177 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^x_p, where x_p and y_p are the coordinates of peak p.

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A258178 Sum over all Dyck paths of semilength n of products over all peaks p of x_p^2, where x_p is the x-coordinate of peak p.

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A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

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A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

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Maple

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A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

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