A006319 -id:A006319 - cp's OEIS frontend

A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 0, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 0, 206098, 164512, 89898, 39152, 14002, 4080, 938, 160, 18, 1

Offset: 0

Philippe Deléham, Sep 18 2006

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . Inverse is Riordan array (1, x*(1-x)/(1+x)).

T(n, r) gives the number of [0,r]-covering hierarchies with n segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014

			Triangle begins:
  1;
  0,    1:
  0,    2,    1;
  0,    6,    4,    1;
  0,   22,   16,    6,    1;
  0,   90,   68,   30,    8,   1;
  0,  394,  304,  146,   48,  10,  1;
  0, 1806, 1412,  714,  264,  70, 12,  1;
  0, 8558, 6752, 3534, 1408, 430, 96, 14, 1;
Production matrix is:
  0...1
  0...2...1
  0...2...2...1
  0...2...2...2...1
  0...2...2...2...2...1
  0...2...2...2...2...2...1
  0...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...1
  0...2...2...2...2...2...2...2...2...1
  ... - _Philippe Deléham_, Feb 09 2014

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.

Another version : A080247, A080245, A033877.
Columns: A000007, A006318, A006319, A006320, A006321.
Diagonals: A000012, A005843, A054000.
Sums include: A001003 (row and alternating sign), A006603 (diagonal).
Cf. A103885.

function T(n,k) // T = A122538
  if k eq 0 then return 0^n;
  elif k eq n then return 1;
  else return T(n-1,k-1) + T(n-1,k) + T(n,k+1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 27 2024

T[n_, n_]= 1; T[, 0]= 0; T[n, k_]:= T[n, k]= T[n-1, k-1] + T[n-1, k] + T[n, k+1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 13 2019 *)

def A122538_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
for n in (0..12): print(A122538_row(n)) # Peter Luschny, Mar 16 2016

T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1) if k > 0, with T(n, 0) = 0^n, and T(n, n) = 1.
Sum_{k=0..n} T(n, k) = A001003(n).
From G. C. Greubel, Oct 27 2024: (Start)
T(2*n, n) = A103885(n).
Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)

A129180 Total area below all Schroeder paths of semilength n.

Original entry on oeis.org

0, 1, 11, 85, 583, 3785, 23843, 147437, 900559, 5453457, 32816315, 196531781, 1172634391, 6976059865, 41401814099, 245230349021, 1450162049695, 8563622372129, 50510963880299, 297627067200821, 1752169739791591, 10307304302433513, 60592569330907523

Offset: 0

Emeric Deutsch, Apr 08 2007

nonn

A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis.

			a(2) = 11 because the areas below the Schroeder paths HH, HUD, UDH, UDUD, UHD and UUDD are 0,1,1,2,3 and 4, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Lara Bossinger and Martina Lanini, Following Schubert varieties under Feigin's degeneration of the flag variety, arXiv:1802.04320 [math.RT], 2018.

Cf. A002315, A006319, A129179.

g:=(1+z)*(1-z-sqrt(1-6*z+z^2))^2/4/z/(1-6*z+z^2): gser:=series(g,z=0,30): seq(coeff(gser,z,n),n=0..24);

CoefficientList[Series[(1 + x)*(1 - x - Sqrt[1 - 6*x + x^2])^2/(4*x*(1 - 6*x + x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 03 2016 *)

a(n):=sum((((sqrt(2)+1)^(2*k+1)-(1-sqrt(2))^(2*k)*sqrt(2)+(1-sqrt(2))^(2*k))*sum(binomial(n+1-k,i+2)*binomial(n-k+i,i),i,0,n-k+1))/(n-k+1),k,0,n); /* Vladimir Kruchinin, Mar 02 2016 */

a(n) = Sum_{k=0..n^2} k * A129179(n,k).
G.f.: (1+z)*(1-z-sqrt(1-6*z+z^2))^2/(4*z*(1-6*z+z^2)) (obtained by computing (dG/dt)_{t=1} where G=G(t,z) is defined by G(t,z) = 1+z*G(t,z)+t*z*G(t,t^2*z)G(t,z); see A129179).
a(n) = Sum_{k=0..n} 2*A002315(k)*(Sum_{i=0..n-k+1} binomial(n+1-k,i+2)*binomial(n-k+i,i))/(n-k+1). - Vladimir Kruchinin, Mar 02 2016
a(n) ~ 1/2 * (1+sqrt(2))^(2*n+1). - Vaclav Kotesovec, Mar 03 2016
D-finite with recurrence -(n+1)*(2*n-5)*a(n) +3*(4*n+1)*(2*n-5)*a(n-1) +(-76*n^2+228*n-89)*a(n-2) +3*(2*n-1)*(4*n-13)*a(n-3) -(2*n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A370479 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/2) / (1-x))^2 )^2.

Original entry on oeis.org

1, 2, 9, 40, 184, 872, 4232, 20936, 105208, 535624, 2757000, 14324456, 75028152, 395750568, 2100380424, 11208429960, 60103977976, 323708642952, 1750294676744, 9497584905128, 51703651336888, 282302043458536, 1545558070957960, 8482843567140680

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A045623, A370477.

Cf. A006319, A370480.

my(N=30, x='x+O('x^N)); Vec((1+x*((1-x-sqrt(1-6*x+x^2))/(2*x))^2)^2)

a(n, r=2, s=2, t=2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: B(x)^2 where B(x) is the g.f. of A006319.

a(n) = 2 * Sum_{k=0..n} binomial(2*k+2,k) * binomial(n+k-1,n-k)/(2*k+2).

A370480 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/3) / (1-x))^2 )^3.

Original entry on oeis.org

1, 3, 15, 73, 360, 1800, 9112, 46632, 240936, 1255336, 6589080, 34811784, 184990568, 988156872, 5303039256, 28579068520, 154605138984, 839272725864, 4570409517848, 24961191298248, 136688674353000, 750355591919240, 4128471397725336, 22762905189252264

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A058396, A370478.

Cf. A006319, A370479.

my(N=30, x='x+O('x^N)); Vec((1+x*((1-x-sqrt(1-6*x+x^2))/(2*x))^2)^3)

a(n, r=3, s=2, t=2, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: B(x)^3 where B(x) is the g.f. of A006319.

a(n) = 3 * Sum_{k=0..n} binomial(2*k+3,k) * binomial(n+k-1,n-k)/(2*k+3).

A382920 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^2 )^3.

Original entry on oeis.org

1, 3, 21, 160, 1320, 11511, 104451, 976317, 9337182, 90937403, 898861308, 8994246132, 90932043400, 927452701605, 9531607969788, 98609173435172, 1026121044859890, 10733030463200814, 112783955395845926, 1190060614961391945, 12604133970419399208, 133945684546835994915

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A006319, A382918.

Cf. A382916.

a(n, r=3, s=2, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(4/3) / (1-x)^2 )^3.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A382916.

A125190 Number of ascents in all Schroeder paths of length 2n.

Original entry on oeis.org

0, 1, 6, 32, 170, 912, 4942, 27008, 148626, 822560, 4573910, 25534368, 143027898, 803467056, 4524812190, 25537728000, 144411206178, 818017823808, 4640757865126, 26364054632480, 149959897539018, 853941394691792, 4867745532495086, 27773897706129792

Offset: 0

Emeric Deutsch, Dec 20 2006

nonn

A Schroeder path of length 2n is a lattice path in the first quadrant, from the origin to the point (2n, 0) and consisting of steps U = (1, 1), D = (1, -1) and H = (2, 0); an ascent in a Schroeder path is a maximal strings of U steps.

a(n) is the number of points at L1 distance n - 2 from any point in Z^n, for n >= 2. - Shel Kaphan, Mar 24 2023

			a(2) = 6 because the Schroeder paths of length 4 are HH, H(U)D, (U)DH, (U)D(U)D, (U)HD and (UU)DD, having a total of 6 ascents (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 0..400
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Cf. A006319, A090981, A008288, A026002, A190666.

-2-diagonal of A266213 for n>=1.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R*(1+z*R)/sqrt(1-6*z+z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..25);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<3, [0,1,6][n+1], ((204*n^2-411*n+198)*a(n-1)
       +(-34*n^2+68*n+96)*a(n-2) +(3*n-12)*a(n-3))/(2*n*(17*n-26)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 20 2012

CoefficientList[Series[x*(1-x-Sqrt[1-6*x+x^2])/(2*x)*(1+x*(1-x-Sqrt[1-6*x+x^2])/(2*x))/Sqrt[1-6*x+x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
a[n_] := Sum[ Binomial[n+1, n-i-1]*Binomial[n+i, n], {i, 0, n-1}]; (* or *) a[n_] := Hypergeometric2F1[1-n, 1+n, 3, -1]*n*(n+1)/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 05 2013, after Vladimir Kruchinin *)

A125190 = lambda n : (n^2+n)*hypergeometric([1-n, n+1], [3], -1)/2
[round(A125190(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 17 2014

a(n) = Sum_{k=0..n} k * A090981(n, k).
G.f.: z*R*(1 + z*R)/sqrt(1 - 6*z + z^2), where R = 1 + z*R + z*R^2, i.e., R = (1 - z -sqrt(1 - 6*z + z^2))/(2*z).
D-finite Recurrence: 2*n*(17*n - 26)*a(n) = 3*(68*n^2 - 137*n + 66)*a(n-1) - 2*(17*n^2 - 34*n - 48)*a(n-2) + 3*(n - 4)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(-3/4)*(3 + 2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2012
a(n) = Sum_{i=0..n-1} binomial(n+1, n-i-1) * binomial(n+i, n). - Vladimir Kruchinin, Feb 05 2013
a(n) = (n*(n+1)/2)*hypergeometric([1-n, n+1], [3], -1). - Peter Luschny, Sep 17 2014
a(n) = A026002(n) - A190666(n-2) for n >= 2. - Shel Kaphan, Mar 24 2023
a(n) = ((n+1)/2) * A006319(n-1). - Vladimir Kruchinin, Apr 27 2024

A167656 Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 16, 14, 4, 22, 68, 78, 40, 8, 90, 304, 410, 284, 104, 16, 394, 1412, 2122, 1776, 896, 256, 32, 1806, 6752, 10966, 10468, 6496, 2592, 608, 64, 8558, 33028, 56870, 59832, 43016, 21376, 7072, 1408, 128, 41586, 164512, 296498, 336252

Offset: 0

Philippe Deléham, Nov 08 2009

			Triangle begins:
1 ;
1,1 ;
2,4,2 ;
6,16,14,4 ;
22,68,78,40,8 ;
90,304,410,284,104,16 ;
...

Cf. A132372, A155069, A006319, A011782.

T(n,0) = T(n-1,0) + T(n-1,1).
T(n,1) = T(n-1,0) + 3*T(n-1,1) + T(n-1,2).
T(n,k) = 2*T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k>1.

A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 3, 4, 7, 14, 27, 48, 84, 152, 284, 532, 987, 1826, 3401, 6384, 12024, 22656, 42728, 80780, 153151, 290970, 553601, 1054688, 2012373, 3845646, 7359345, 14100692, 27048061, 51941850, 99855389, 192163904, 370159216, 713672568, 1377168108, 2659729380

Offset: 0

Seiichi Manyama, Oct 15 2023

nonn

Partial sums give A023426.

Cf. A002026, A006319, A052702.

A361229 := proc(n)
    add(binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1),k=0..floor(n/4)) ;
end proc:
seq(A361229(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

a(n) = sum(k=0, n\4, binomial(n-2*k-1, n-4*k)*binomial(2*k, k)/(k+1));

G.f.: A(x) = 2*(1-x) / (1-x+sqrt((1-x)^2-4*x^4)).
a(n) = Sum_{k=0..floor(n/4)} binomial(n-2*k-1,n-4*k) * binomial(2*k,k) / (k+1).
D-finite with recurrence (n+4)*a(n) +(-3*n-7)*a(n-1) +(3*n+2)*a(n-2) +(-n+1)*a(n-3) +4*(-n+2)*a(n-4) +4*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023

A382918 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^2 )^2.

Original entry on oeis.org

1, 2, 11, 64, 401, 2652, 18241, 129216, 936469, 6911238, 51764834, 392494366, 3006851913, 23238830982, 180974578418, 1418728452902, 11186978492689, 88668723061112, 706042492550773, 5645331629000370, 45307653034905824, 364860349786846894, 2947299389835541583

Offset: 0

Seiichi Manyama, Apr 08 2025

nonn

Cf. A006319, A382920.

Cf. A366176.

a(n, r=2, s=2, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(3/2) / (1-x)^2 )^2.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^2, where B(x) is the g.f. of A366176.

A052736 E.g.f. [1 -3x -sqrt(1-6x+x^2) -x*(1-x-sqrt(1-6x+x^2)) ]/2.

Original entry on oeis.org

0, 0, 2, 24, 384, 8160, 218880, 7116480, 272240640, 11985200640, 596981145600, 33195609216000, 2038500521164800, 137021183973273600, 10006412139653529600, 788930789450259456000, 66790064645111808000000, 6042970648669883056128000, 581917311773908793819136000, 59423732260666221275283456000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 692

spec := [S,{B=Prod(Z,C),S=Prod(C,C),C=Union(B,S,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20); # end of program
0, seq(simplify( n!*(n-1)*hypergeom([n, 2-n],[3],-1) ), n=1..20);  # Mark van Hoeij, May 29 2013

CoefficientList[Series[1/2-3/2*x-1/2*(1-6*x+x^2)^(1/2)-(1/2-1/2*x-1/2*(1-6*x+x^2)^(1/2))*x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

x='x+O('x^66); concat([0,0], Vec( serlaplace( 1/2-3/2*x -1/2*(1-6*x+x^2)^(1/2) -(1/2-1/2*x-1/2*(1-6*x+x^2)^(1/2))*x))) \\ Joerg Arndt, May 29 2013

D-finite with recurrence: a(1)=0; a(2)=2; a(3)=24; (-n^3-n^2+4*n+4)*a(n) +(-6+7*n^2+11*n)*a(n+1) +(-7*n-10)*a(n+2) +a(n+3) =0.
a(n) ~ (2-sqrt(2))*sqrt(3*sqrt(2)-4)*n^(n-1)*(3+2*sqrt(2))^n/exp(n). - Vaclav Kotesovec, Oct 05 2013
Conjecture: a(n) = n!*A006319(n-1). - R. J. Mathar, Oct 16 2013

cp's OEIS Frontend

A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

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Magma

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Sage

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A129180 Total area below all Schroeder paths of semilength n.

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Mathematica

Maxima

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A370479 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/2) / (1-x))^2 )^2.

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PARI

PARI

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A370480 G.f. satisfies A(x) = ( 1 + x * (A(x)^(1/3) / (1-x))^2 )^3.

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PARI

PARI

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A382920 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) / (1-x)^2 )^3.

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PARI

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A125190 Number of ascents in all Schroeder paths of length 2n.

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Maple

Mathematica

Sage

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A167656 Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,1,2,1,2,1,2,1,2,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A361229 G.f. A(x) satisfies A(x) = 1 + x^4 * (A(x) / (1 - x))^2.

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Maple