A054727 -id:A054727 - cp's OEIS frontend

A381773 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^3 ) )^(1/3), where C(x) is the g.f. of A000108.

1, 2, 15, 157, 1913, 25427, 357546, 5229980, 78765793, 1213181593, 19021747383, 302595975502, 4871780511910, 79232327379407, 1299767617080662, 21481625997258747, 357350097625089497, 5978708468143961925, 100537111802285439375, 1698302173359384479307

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381772, A381774, A381775.
Cf. A212071, A381782, A381783.
Cf. A234461, A381779, A381780, A381786.
Cf. A000108.

my(N=30, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^3)/x)^(1/3))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * C(x*A(x)^3).

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

A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 83, 727, 6940, 70058, 735502, 7949031, 87851819, 988307647, 11279719247, 130286197186, 1520108988221, 17889102534329, 212095541328931, 2531001870925559, 30376237591559863, 366417240105654587, 4440000077166319993, 54020150448778625847, 659665548217188211288

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381773, A381774, A381775.
Cf. A007852, A069271, A381780.
Cf. A000108, A274052.

my(N=30, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^2)/x)^(1/2))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * C(x*A(x)^2).

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

A381774 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 19, 255, 3995, 68344, 1237526, 23316295, 452385355, 8977539540, 181374792040, 3718002102747, 77138798530854, 1616741658725930, 34179703551312530, 728019711835819493, 15608122038151106507, 336551042553481867640, 7293934071668996347055

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381772, A381773, A381775.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^4)/x)^(1/4))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^4) * C(x*A(x)^4).

a(n) = Sum_{k=0..n} binomial(4*n+2*k+1,k) * binomial(4*n+1,n-k)/(4*n+2*k+1).

A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Original entry on oeis.org

1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318

Offset: 0

Janos A. Csirik, Apr 21 2000

sign

a(150) = -1, a(n) > 0 for 0<=n<=149.

a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054727, A054729.

a1617[n_] := If[n<1, 0, 1+Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 - Count[(#^2+1)/n& /@ Range[n], ?IntegerQ]/4];
seq[n_] := Module[{inv, bnd}, inv = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n]] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n && inv[[g+1]] == -1, inv[[g+1]] = k]]; inv];
seq[51] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
seq(n) = {
  my(inv = vector(n+1,g,-1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  return(inv);
};
seq(51)  \\ Gheorghe Coserea, May 21 2016

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

A118343 Triangle, read by rows, where diagonals are successive self-convolutions of A108447.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 20, 0, 1, 4, 15, 48, 113, 0, 1, 5, 22, 85, 282, 688, 0, 1, 6, 30, 132, 519, 1762, 4404, 0, 1, 7, 39, 190, 837, 3330, 11488, 29219, 0, 1, 8, 49, 260, 1250, 5516, 22135, 77270, 199140, 0, 1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 0

Offset: 0

Paul D. Hanna, Apr 26 2006

A108447 equals the central terms of pendular triangle A118340 and the diagonals of this triangle form the semi-diagonals of the triangle A118340. Row sums equal A054727, the number of forests of rooted trees with n nodes on a circle without crossing edges.

			Show: T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n,k-1) + T(n+1,k-1)
at n=8,k=4: T(8,4) = T(7,4) - T(7,3) + T(8,3) + T(9,3)
or 837 = 519 - 132 + 190 + 260.
Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  4,   0;
  1, 3,  9,  20,    0;
  1, 4, 15,  48,  113,    0;
  1, 5, 22,  85,  282,  688,     0;
  1, 6, 30, 132,  519, 1762,  4404,      0;
  1, 7, 39, 190,  837, 3330, 11488,  29219,      0;
  1, 8, 49, 260, 1250, 5516, 22135,  77270, 199140,       0;
  1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A054727 (row sums), A108447, A118340.

T:= proc(n, k) option remember;
      if k<0 or  k>n then 0;
    elif k=0 then 1;
    elif k=n then 0;
    else T(n-1, k) -T(n-1, k-1) +T(n, k-1) +T(n+1, k-1);
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 17 2021

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -T[n-1, k-1] +T[n, k-1] +T[n+1, k-1] ]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2021 *)

{T(n,k)=polcoeff((serreverse(x*(1-x+sqrt((1-x)*(1-5*x)+x*O(x^k)))/2/(1-x))/x)^(n-k),k)}

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (k==n): return 0
    else: return T(n-1, k) -T(n-1, k-1) +T(n, k-1) +T(n+1, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 17 2021

Since g.f. G=G(x) of A108447 satisfies: G = 1 - x*G + x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [2*T(n-1+j,k-j) - T(n-2+j,k-j)] for n>k>=0.

Sum_{k=0..n} T(n,k) = [n=0] + A054727(n) = [n=0] + Sum_{j=1..n} binomial(n, j-1)*binomial(3*n-2*j-1, n-j)/(2*n-j). - G. C. Greubel, Mar 17 2021

A303730 Number of noncrossing path sets on n nodes with each path having at least two nodes.

Original entry on oeis.org

1, 0, 1, 3, 10, 35, 128, 483, 1866, 7344, 29342, 118701, 485249, 2001467, 8319019, 34810084, 146519286, 619939204, 2635257950, 11248889770, 48198305528, 207222648334, 893704746508, 3865335575201, 16761606193951, 72860178774410, 317418310631983, 1385703968792040

Offset: 0

Andrew Howroyd, Apr 29 2018

nonn

Paths are constructed using noncrossing line segments between the vertices of a regular n-gon. Isolated vertices are not allowed.

A noncrossing path set is a noncrossing forest (A054727) where each tree is restricted to being a path.

			Case n=3: There are 3 possibilities:
.
     o       o       o
    /         \     / \
   o---o   o---o   o   o
.
Case n=4: There are 10 possibilities:
.
   o   o   o   o   o---o   o---o   o---o
   |   |   |   |   |       |   |       |
   o   o   o---o   o---o   o   o   o---o
.
   o---o   o---o   o---o   o   o   o   o
             /       \     | / |   | \ |
   o---o   o---o   o---o   o   o   o   o
.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A054727, A303729.

InverseSeries[x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O[x]^30, x] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Jul 03 2018, from PARI *)

Vec(serreverse(x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3) + O(x^30)))

G.f.: G(x)/x where G(x) is the reversion of x*(1 - 2*x)^2/(1 - 4*x + 5*x^2 - x^3).

A381882 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * C(x)) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 4, 24, 175, 1428, 12525, 115468, 1103777, 10844715, 108860766, 1111722956, 11514401451, 120666441067, 1277161022725, 13633269293868, 146606818816257, 1586739194404521, 17271207134469417, 188942438655850740, 2076317084779878706, 22909617070555385010

Offset: 0

Seiichi Manyama, Mar 09 2025

nonn

Cf. A054727, A381881.

Cf. A000108, A381880.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3*(1-sqrt(1-4*x))/(2*x)))/x)

G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * C(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+3,n-k)/(n+2*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/2, 1+n/2, -n], [2 + n, 4 + 2*n], -4)/(1 + n). - Stefano Spezia, Mar 09 2025

A094021 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 12, 14, 6, 1, 55, 75, 40, 10, 1, 273, 429, 275, 90, 15, 1, 1428, 2548, 1911, 770, 175, 21, 1, 7752, 15504, 13328, 6370, 1820, 308, 28, 1, 43263, 95931, 93024, 51408, 17640, 3822, 504, 36, 1, 246675, 600875, 648945, 406980, 162792, 42840

Offset: 1

Emeric Deutsch, May 31 2004

			From _Andrew Howroyd_, Nov 17 2017: (Start)
Triangle begins:
     1;
     1,     1;
     3,     3,     1;
    12,    14,     6,    1;
    55,    75,    40,   10,    1;
   273,   429,   275,   90,   15,   1;
  1428,  2548,  1911,  770,  175,  21,  1;
  7752, 15504, 13328, 6370, 1820, 308, 28, 1;
(End)
T(3,2)=3 because, with A,B,C denoting the vertices of a triangle, we have the 2-component forests (A,BC), (B,CA) and (C,AB).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Columns k=1..2 are A001764, A026004.
Row sums are A054727.
Cf. A000108.

T:=proc(n,k) if k<=n then binomial(n,k-1)*binomial(3*n-2*k-1,n-k)/(2*n-k) else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..11);

T[n_, k_] := If[k <= n, Binomial[n, k-1]*Binomial[3n-2k-1, n-k]/(2n-k), 0];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 06 2018 *)

T(n,k)=binomial(n, k-1)*binomial(3*n-2*k-1, n-k)/(2*n-k);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

T(n, k) = binomial(n, k-1)*binomial(3n-2k-1, n-k)/(2n-k).
G.f.: G=G(t, z) satisfies G^3+(t^3*z^2-t^2*z-3)G^2+(t^2*z+3)G-1=0.
From Peter Bala, Nov 07 2015: (Start)
O.g.f. A(x,t) = revert( x/((1 + x*t)*C(x)) ) with respect to x, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
Row sums are A054727. (End)

A381775 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^6 ) )^(1/6), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 27, 523, 11871, 294668, 7747698, 212054604, 5978347887, 172421233231, 5063192676597, 150872475295522, 4550458484780442, 138652322209300991, 4261638256558924407, 131973650298641750844, 4113788296015093994719, 128973000885015536107140

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381772, A381773, A381774.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^6)/x)^(1/6))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^6) * C(x*A(x)^6).
a(n) = Sum_{k=0..n} binomial(6*n+2*k+1,k) * binomial(6*n+1,n-k)/(6*n+2*k+1).
a(n) = binomial(1 + 6*n, n)*hypergeom([-n, 1/2+3*n, 1+3*n], [2+5*n, 2+6*n], -4)/(1 + 6*n). - Stefano Spezia, Mar 07 2025

A381778 G.f. A(x) satisfies A(x) = (1 + xA(x)) C(x*A(x)^2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 9, 60, 474, 4105, 37681, 360122, 3545320, 35705553, 366126614, 3809497971, 40119258081, 426829897847, 4580629916321, 49527776299522, 539025763347730, 5900193301962178, 64913644702760248, 717433047054489969, 7961616716665723173, 88679610762886209459

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A381779.

Cf. A000108.

a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(n+k+1, n-k)/(n+3*k+1));

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

cp's OEIS Frontend

A381773 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^3 ) )^(1/3), where C(x) is the g.f. of A000108.

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A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

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A381774 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

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A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

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A118343 Triangle, read by rows, where diagonals are successive self-convolutions of A108447.

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A303730 Number of noncrossing path sets on n nodes with each path having at least two nodes.

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A381882 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * C(x)) ), where C(x) is the g.f. of A000108.

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A094021 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k components (1<=k<=n).

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A381778 G.f. A(x) satisfies A(x) = (1 + xA(x)) C(x*A(x)^2), where C(x) is the g.f. of A000108.