A144097 -id:A144097 - cp's OEIS frontend

A363006 a(n) = 1/((d-1)n + 1)Sum_{i=0..n} binomial((d - 1)n+1, n-i) binomial((d-1)*n+i, i), with d = 6.

1, 2, 22, 342, 6202, 122762, 2571326, 56031470, 1257199154, 28849835538, 673953255142, 15973925161030, 383186776643946, 9285457458463770, 226959074854361742, 5588974707042304222, 138529985051020001634, 3453373395317346136610, 86526667346028323084726, 2177844556015530807952438

Offset: 0

Michael De Vlieger, May 16 2023

nonn

See Yang-Jiang paper, related to large Schröder numbers, which correspond to the formula in the Name, instead with d=2.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. See p. 145. (in Mandarin.)

Cf. A006318 (d=2), A027307 (d=3), A144097 (d=4), A260332 (d=5).

Cf. A217364, A363305, A364195, A364196.

With[{d = 6}, Table[(1/((d - 1) n + 1)) Sum[Binomial[(d - 1) n + 1, n - i] Binomial[(d - 1) n + i, i], {i, 0, n}], {n, 0, 12}] ]

a(n) = my(d=6); sum(i=0, n, binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i))/((d-1)*n + 1); \\ Michel Marcus, May 16 2023

G.f. satisfies A(x) = 1 + x * A(x)^5 * (1 + A(x)). - Seiichi Manyama, May 29 2023
From Seiichi Manyama, Aug 09 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(5*n,k-1) for n > 0.  (End)

A371675 G.f. satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 32, 324, 3696, 45316, 583152, 7769348, 106250144, 1482925956, 21037812352, 302478044996, 4397824031376, 64549296707460, 955150116019920, 14233474784850948, 213417133281087040, 3217460713030341892, 48741781832765496288, 741606216370357708612

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Column k=2 of A378238.
Cf. A006319, A032349, A371676. A371677, A371678.
Cf. A144097, A371679.

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

G.f. satisfies A(x) = ( 1 + x * A(x)^(3/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A144097.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).
a(n) ~ sqrt((88 + 161*sqrt(2/5))/Pi) * (223 + 70*sqrt(10))^n / (n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

A108424 Number of paths from (0,0) to (3n,0) that stay in the first quadrant, consist of steps u=(2,1), U=(1,2), or d=(1,-1) and do not touch the x-axis, except at the endpoints.

Original entry on oeis.org

2, 6, 34, 238, 1858, 15510, 135490, 1223134, 11320066, 106830502, 1024144482, 9945711566, 97634828354, 967298498358, 9659274283650, 97119829841854, 982391779220482, 9990160542904134, 102074758837531810, 1047391288012377774, 10788532748880319298

Offset: 1

Emeric Deutsch, Jun 03 2005

nonn

These are the large nu-Schröder numbers with nu=NE(NEE)^(n-1). - Matias von Bell, Jun 02 2021

			a(2) = 6 because we have uudd, uUddd, Ududd, UdUddd, Uuddd and UUdddd.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
M. von Bell and M. Yip, Schröder combinatorics and nu-associahedra, arXiv:2006.09804 [math.CO], 2020.

Cf. A027307, A032349, A344553.

Cf. A006318 (d = 2, signed version at d = 0), A027307 (d = 3), A144097 (d = 4), A260332 (d = 5, conjecturally), A363006 (d = 6).

A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: G:=z*A+z*A^2: Gser:=series(G,z=0,28): seq(coeff(Gser,z^n),n=1..25);
a:=proc(n) if n=1 then 2 else (n*2^n*binomial(2*n,n)/((2*n-1)*(n+1)))*sum(binomial(n-1,j)^2/2^j/binomial(n+j+1,j),j=0..n-1) fi end: seq(a(n),n=1..19);
# Alternative:
a := n -> 2*binomial(3*n - 2, 2*n - 1)*hypergeom([2 - 2*n, 1 - n], [2 - 3*n], -1)/n:
seq(simplify(a(n)), n = 1..21); # Peter Luschny, Jun 14 2021

Table[(n*2^n*Binomial[2*n,n]/((2n-1)*(n+1))) * Sum[(Binomial[n-1,j])^2/ (2^j * Binomial[n+j+1,j]), {j,0,n-1}], {n,1,20}] (* Vaclav Kotesovec, Oct 17 2012 *)

a(n) = A027307(n-1) + A032349(n).
G.f.: z*A+z*A^2, where A=1+z*A^2+z*A^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3.
a(n) = (n*2^n*C(2*n, n)/((2n-1)(n+1))) * Sum_{j=0..n-1} (C(n-1, j))^2 / (2^j*C(n+j+1,j)).
Recurrence: n*(2*n-1)*a(n) = 3*(6*n^2-10*n+3)*a(n-1) + (46*n^2-227*n+279)*a(n-2) + 2*(n-3)*(2*n-7)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(30*sqrt(5) - 50)*((11 + 5*sqrt(5))/2)^n/(20*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n) = Sum_{i=0..n} (2*n+i-2)!/((n-i)!*(n+i-1)!*i!), n>0. - Vladimir Kruchinin, Feb 16 2013
From Matias von Bell, Jun 02 2021: (Start)
a(n) = 2*Sum_{i>=0} (1/n)*binomial(2*n-2,i)*binomial(3*n-2-i,2*n-1).
a(n) = 2*A344553(n). (End)
a(n) = 2*binomial(3*n - 2, 2*n - 1)*hypergeom([2 - 2*n, 1 - n], [2 - 3*n], -1) / n. - Peter Luschny, Jun 14 2021
From Peter Bala, Jun 17 2023: (Start)
a(n) = (-1)^(n+1) * (1/((d-1)*n + 1))*Sum_{i = 0..n} binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i), with d = -1.
P-recursive: n*(2*n - 1)*(5*n - 8)*a(n) = (110*n^3 - 396*n^2 + 445*n - 150)*a(n-1) + (n - 2)*(2*n - 5)*(5*n - 3)*a(n-2) with a(1) = 2 and a(2) = 6.
The g.f. A(x) = 2*x + 6*x^2 + 34*x^3 + .... Then 1/(1 - A(x)) = 1 + 2*x + 10*x^2 + 66*x^3 + .. is the g.f. of A027307.
(1/x) * the series reversion of x*(1 - A(x)) = 1 + 2*x + 14*x^2 + 134*x^3 + ... is the g.f. of A144097.
(1/x) * the series reversion of x/(1 - A(x)) = 1 - 2*x - 2*x^2 - 6*x^3 - 22*x^4 - 90*x^5 - ... =  1 - x - x*S(x), where S(x) is the g.f. of A006318. (End)

A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 6, 2, 1, 2, 10, 22, 2, 1, 2, 14, 66, 90, 2, 1, 2, 18, 134, 498, 394, 2, 1, 2, 22, 226, 1482, 4066, 1806, 2, 1, 2, 26, 342, 3298, 17818, 34970, 8558, 2, 1, 2, 30, 482, 6202, 52450, 226214, 312066, 41586, 2, 1, 2, 34, 646, 10450, 122762, 881970, 2984206, 2862562, 206098, 2

Offset: 0

Seiichi Manyama, Jul 25 2020

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  2,   2,    2,     2,     2,      2, ...
  2,   6,   10,    14,    18,     22, ...
  2,  22,   66,   134,   226,    342, ...
  2,  90,  498,  1482,  3298,   6202, ...
  2, 394, 4066, 17818, 52450, 122762, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give A040000, A006318, A027307, A144097.
If Michael D. Weiner's conjecture on A260332 is correct, column 4 is A260332 for n > 0.
Main diagonal gives A336537.
Cf. A336521, A336574. A336575.

T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n+j+1, n]/(k*n+j+1), {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

T(n, k) = sum(j=0, n, binomial(k*n+1, j)*binomial((k+1)*n-j, n-j))/(k*n+1);

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (1 + A_k(x)).
T(n,k) = (1/n) * Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = binomial(1+k*n, n)*hypergeom([-n, 1+k*n], [2+(k-1)*n], -1)/(1 + k*n) for k > 0. - Stefano Spezia, Aug 09 2025

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+r+k,n)/(3n+r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

Columns k=0..3 give A000007, A144097, A371675, A365843.
T(n,n) gives 1/4 * A370102(n) for n > 0.
Cf. A033877, A071949, A378236, A378237, A378239, A378240.

T(n, k, t=3, u=1) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

Original entry on oeis.org

1, 2, 18, 234, 3570, 59586, 1053570, 19392490, 367677090, 7131417282, 140834140722, 2822214963882, 57243994984722, 1172991472484610, 24245748916730658, 504935751379031082, 10584721220759172162, 223163804001804187266, 4729176407109705542994, 100676187744957784842090

Offset: 0

Seiichi Manyama, Jul 13 2023

Alois P. Heinz, Table of n, a(n) for n = 0..737 (first 50 terms from Christian N. Hofmann)

Cf. A144097, A153231, A363311.

a:= n-> sum(binomial(n, k)*binomial(3*n+3*k+1, n)/(3*n+3*k+1), k=0..n):
seq(a(n), n=0..49); # Christian N. Hofmann, Jul 14 2023

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

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

A364396 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)).

Original entry on oeis.org

1, 2, -10, 86, -902, 10506, -130594, 1697006, -22774094, 313205522, -4391039930, 62522730310, -901680559574, 13143551082138, -193339856081490, 2866341942620382, -42784807130635678, 642457682754511906, -9698259831536382826, 147091417979841002294

Offset: 0

Seiichi Manyama, Jul 22 2023

sign

Cf. A112478, A364394, A364398.

Cf. A144097.

a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(3*n+k-2, n-1))/n);

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A144097.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(3*n+k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n+1)*4^(-n)*27^n*n^(-3/2)*2F1([-n, 3*n-1], [2*n], -1), with c = 1/(3*sqrt(3*Pi)). - Stefano Spezia, Oct 21 2023

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).

Original entry on oeis.org

1, 2, 18, 222, 3166, 49098, 804138, 13686198, 239671590, 4290463698, 78160665666, 1444298971662, 27005948771886, 510024567278234, 9714561608833242, 186403770207998310, 3599812021110287862, 69914211761486437026, 1364692279095996581490

Offset: 0

Seiichi Manyama, Aug 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..757

Cf. A025192, A107841, A235347, A364826, A364827.

Cf. A144097, A243659.

A364825 := proc(n)
    (-1)^n*add( (-3)^k*binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364825(n),n=0..80); # R. J. Mathar, Aug 10 2023

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

a(n) = (-1)^n * Sum_{k=0..n} (-3)^k * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +2079*n*(3*n-1)*(3*n+1)*a(n) +(-347173*n^3 +395007*n^2 -41030*n -43092)*a(n-1) +18*(-59207*n^3 +325826*n^2 -590255*n +352406)*a(n-2) +3*(-3299*n^3 +35998*n^2 -125399*n +141144)*a(n-3) +9*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 10 2023

A257995 Forests of binary shrubs on 3n vertices avoiding 321.

Original entry on oeis.org

1, 2, 37, 866, 23285, 679606, 20931998, 669688835, 22040134327, 741386199872, 25376258521393, 880977739374392, 30946637156662975, 1097929752363923490, 39284677690031136567, 1415992852373003788459

Offset: 0

Manda Riehl, May 15 2015

nonn

We define a shrub as a rooted, ordered tree with the only vertices being the root and leaves. We then label our shrubs' vertices with integers such that each child has a larger label than its parent. We associate a permutation to a tree by reading the labels from left to right by levels, starting with the root. A forest is an ordered collection of trees where all vertices in the forest have distinct labels. We associate a permutation to a forest by reading the permutation associated to each tree and then concatenating. We then enumerate labeled forests of binary shrubs whose associated permutation avoids 321.

David Bevan, Table of n, a(n) for n = 0..993
D Bevan, D Levin, P Nugent, J Pantone, L Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036, 2015
M. Riehl, Forests of binary shrubs avoiding patterns of length 3

A001764, A002293, A060941 and A144097 enumerate binary shrubs avoiding other patterns of length 3.

gf := RootOf(_Z^10*z^10+18*_Z^9*z^9+123*_Z^8*z^8+(-3*z^8+420*z^7+54*z^6)*_Z^7+(-36*z^7+751*z^6+486*z^5)*_Z^6+(-138*z^6+354*z^5+1053*z^4)*_Z^5+(3*z^6-228*z^5-213*z^4+162*z^3+729*z^2)*_Z^4+(18*z^5-215*z^4+2*z^3-360*z^2)*_Z^3+(15*z^4+24*z^3-71*z^2-54*z)*_Z^2+(-z^4+24*z^3-8*z^2+54*z-1)*_Z+4*z^2+4*z+1)^(1/2):
seq(coeff(series(gf,z,21),z,i),i=0..20);

b[k_]:=k(k+1)/2;n[k_]:=n[k]=Join[{b[k+1],b[k+1]-1},Table[b[i],{i,k,1,-1}],{1}];v[1]={1,0,1};v[k_]:=v[k]=Module[{s=MapIndexed[#1n[First@#2]&,v[k-1]]},Table[Total[If[i>Length@#,0,#[[i]]]&/@s],{i,Length@Last@s}]];a[k_]:=a[k]=Total@v[k];Array[a,20] (* David Bevan, Oct 27 2015 *)

A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

Original entry on oeis.org

1, 2, 26, 482, 10450, 247554, 6208970, 162064322, 4356511138, 119788611458, 3353361311738, 95251219926690, 2738421518770546, 79531905952256642, 2329955712706784682, 68770993359030211458, 2043143866891345880898, 61050342965542475675906

Offset: 0

Seiichi Manyama, Apr 03 2024

nonn

Cf. A006318, A027307, A144097, A260332, A363006.

Cf. A371678.

a(n, r=1, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(6*n,k-1) for n > 0.

cp's OEIS Frontend

A363006 a(n) = 1/((d-1)*n + 1)*Sum_{i=0..n} binomial((d - 1)*n+1, n-i) * binomial((d-1)*n+i, i), with d = 6.

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A371675 G.f. satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(1/2))^2.

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A108424 Number of paths from (0,0) to (3n,0) that stay in the first quadrant, consist of steps u=(2,1), U=(1,2), or d=(1,-1) and do not touch the x-axis, except at the endpoints.

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A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).

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A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+r+k,n)/(3*n+r+k) for k > 0.

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A364167 Expansion of g.f. A(x) satisfying A(x) = 1 + x * A(x)^3 * (1 + A(x)^3).

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A364396 G.f. satisfies A(x) = 1 + x/A(x)^2*(1 + 1/A(x)).

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A364825 G.f. satisfies A(x) = 1 - x*A(x)^3 * (1 - 3*A(x)).

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A363006 a(n) = 1/((d-1)n + 1)Sum_{i=0..n} binomial((d - 1)n+1, n-i) binomial((d-1)*n+i, i), with d = 6.

A336534 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(kn+j+1,n)/(kn+j+1).

A378238 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3n+r+k,n)/(3n+r+k) for k > 0.

A364825 G.f. satisfies A(x) = 1 - xA(x)^3 (1 - 3*A(x)).