A032349 -id:A032349 - cp's OEIS frontend

A102230 Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.

1, 1, 1, 4, 5, 1, 24, 32, 9, 1, 172, 236, 76, 13, 1, 1360, 1896, 656, 136, 17, 1, 11444, 16116, 5828, 1348, 212, 21, 1, 100520, 142544, 53112, 13184, 2376, 304, 25, 1, 911068, 1298524, 494364, 128924, 25436, 3804, 412, 29, 1, 8457504, 12100952

Offset: 0

Paul D. Hanna, Jan 01 2005

Row sums equal A027307; the self-convolution of the row sums form A032349. Column 0 equals A032349 shift right. Column 1 is A102231. This triangle is a variant of A100326.

			This triangle is generated by the recurrence:
T(n,k) = Sum_{i=0..n-k} T(i+1,0)*T(n-i-1,k-1) for n>k>0,
T(n,0) = Sum_{i=0..n-1} (2*i+1)*T(n-1,i) for n>0, with T(0,0)=1.
Rows begin:
[1],
[1,1],
[4,5,1],
[24,32,9,1],
[172,236,76,13,1],
[1360,1896,656,136,17,1],
[11444,16116,5828,1348,212,21,1],
[100520,142544,53112,13184,2376,304,25,1],...
Column 0 is formed from the partial sums of the prior row
after a term-by-term product with the odd numbers:
T(2,0) = 1*T(1,0) + 3*T(1,1) = 1*1 + 3*1 = 4.
T(3,0) = 1*T(2,0) + 3*T(2,1) + 5*T(2,2) = 1*4 + 3*5 + 5*1 = 24.

Cf. A032349, A027307, A102231, A100326.

PARI
```
{T(n,k)=if(n
				
```

G.f.: A(x, y) = (1+x*F(x))/(1-x*y*F(x)) where F(x) is the g.f. of A032349 and satisfies F(x) = (1+x*F(x))^2/(1-x*F(x))^2.

A102231 Column 1 of triangle A102230 and equals the convolution of A032349 with A032349 shift right.

Original entry on oeis.org

0, 1, 5, 32, 236, 1896, 16116, 142544, 1298524, 12100952, 114820964, 1105574400, 10775285836, 106098789832, 1053858546516, 10546951101360, 106249238782652, 1076554249491640, 10964085715303620, 112175072002688480

Offset: 0

Paul D. Hanna, Jan 01 2005

nonn

Cf. A032349, A102230.

{a(n)=local(B=vector(n+1,k,if(k==1,1, sum(i=0,k-2,2^(i+1)*binomial(2*k-2,i)*binomial(k-1,i+1))/(k-1)))); return(polcoeff(Ser(B)^2+x*Ser(B)^4+x*O(x^n),n))}

A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

Original entry on oeis.org

1, 6, 54, 578, 6810, 85278, 1113854, 15004746, 206955378, 2908113974, 41484917958, 599202514578, 8745727050762, 128790559374030, 1911191826600462, 28551332345784730, 429040549473424866, 6480799118506040934, 98349636147075506006, 1498732955394826784226

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Column k=3 of A378238.
Cf. A007297, A365842, A365844, A365845.
Cf. A032349, A365622, A365847.
Cf. A144097.

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

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*(n+1),n-k).
G.f.: B^3, where B is the g.f. of A144097.
a(n) ~ sqrt(8060 + 2651*sqrt(10)) * (223 + 70*sqrt(10))^n / (2 * sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

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

A100324 Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 14, 1, 4, 12, 34, 79, 1, 5, 18, 61, 195, 494, 1, 6, 25, 96, 357, 1230, 3294, 1, 7, 33, 140, 575, 2277, 8246, 22952, 1, 8, 42, 194, 860, 3716, 15372, 57668, 165127, 1, 9, 52, 259, 1224, 5641, 25298, 108018, 415995, 1217270

Offset: 0

Paul D. Hanna, Nov 16 2004

Column k forms the binomial transform of row k in triangle A100326 for k>=0.

			Array, A(n,k), begins as:
  1, 1,  3,  14,   79,   494,  3294, ...;
  1, 2,  7,  34,  195,  1230,  8246, ...;
  1, 3, 12,  61,  357,  2277, 15372, ...;
  1, 4, 18,  96,  575,  3716, 25298, ...;
  1, 5, 25, 140,  860,  5641, 38775, ...;
  1, 6, 33, 194, 1224,  8160, 56695, ...;
  1, 7, 42, 259, 1680, 11396, 80108, ...;
Antidiagonal triangle, T(n,k), begins as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7,  14;
  1, 4, 12,  34,  79;
  1, 5, 18,  61, 195,  494;
  1, 6, 25,  96, 357, 1230, 3294;
  1, 7, 33, 140, 575, 2277, 8246, 22952;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A003169, A032349, A100325, A100326.

f[n_]:= f[n]= If[n<2, 1, If[n==2, 3, ((324*n^2-708*n+360)*f[n-1] - (371*n^2-1831*n+2250)*f[n-2] +(20*n^2-130*n+210)*f[n-3])/(16*n*(2*n -1)) ]]; (* f = A003169 *)
A[n_, k_]:= A[n, k]= If[n==0, f[k], If[k==0, 1, Sum[A[0,k-j]*A[n-1,j], {j,0,k}]]]; (* A = A100324 *)
T[n_, k_]:= A[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 31 2023 *)

{A(n,k)=if(k==0,1,if(n>0,sum(i=0,k,A(0,k-i)*A(n-1,i)), if(k==1,1,if(k==2,3,( (324*k^2-708*k+360)*A(0,k-1)-(371*k^2-1831*k+2250)*A(0,k-2)+(20*k^2-130*k+210)*A(0,k-3))/(16*k*(2*k-1)) )));)}

def f(n): # f = A003169
    if (n<2): return 1
    elif (n==2): return 3
    else: return ((324*n^2-708*n+360)*f(n-1) - (371*n^2-1831*n+2250)*f(n-2) + (20*n^2-130*n+210)*f(n-3))/(16*n*(2*n-1))
@CachedFunction
def A(n, k): # A = 100324
    if (n==0): return f(k)
    elif (k==0): return 1
    else: return sum( A(0,k-j)*A(n-1, j) for j in range(k+1) )
def T(n,k): return A(n-k,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 31 2023

A(n, k) = Sum_{i=0..k} A(0, k-i)*A(n-1, i) for n>0.
A(0, k) = A003169(k+1) = ( (324*k^2-708*k+360)*A(0, k-1) - (371*k^2-1831*k+2250)*A(0, k-2) +(20*k^2-130*k+210)*A(0, k-3) )/(16*k*(2*k-1)) for k>2, with A(0, 0) = A(0, 1)=1, A(0, 2)=3.
A(n, n) = (n+1)*A032349(n+1).
T(n, k) = A(n-k, k) (Antidiagonal triangle).
T(n, n) = A003169(n+1).
Sum_{k=0..n} T(n, k) = A100325(n) (Antidiagonal row sums).

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)

A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

Original entry on oeis.org

1, 8, 96, 1368, 21440, 356968, 6197408, 110947768, 2033381760, 37963483592, 719495148768, 13806129179928, 267693334199616, 5236670783633960, 103227182363423008, 2048451544990578552, 40888361539777714944, 820400146864231266184

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Cf. A066357, A365754, A365846, A365848.
Cf. A032349, A365622, A365843.
Cf. A260332.

a(n) = sum(k=0, n, binomial(4*n+k+3, k)*binomial(4*(n+1), n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(4*(n+1),n-k).

Conjecture: g.f.: B^4, where B is the g.f. of A260332.

A084078 Length of list created by n substitutions k -> Range[-abs(k+1), abs(k-1), 2] starting with {0}.

Original entry on oeis.org

1, 2, 4, 10, 24, 66, 172, 498, 1360, 4066, 11444, 34970, 100520, 312066, 911068, 2862562, 8457504, 26824386, 80006116, 255680170, 768464312, 2471150402, 7474561164, 24161357010, 73473471344, 238552980386, 728745517972

Offset: 0

Wouter Meeussen, May 11 2003

nonn

			{0}, {-1,1}, {0,2,-2,0}, {-1,1,-3,-1,1,-1,1,3,-1,1}

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

Cf. A027307, A084075.

I:=[1,2,4,10]; [n le 4 select I[n] else (6*(35*n^2-125*n+14)*Self(n-1) + (275*n^4 -1870*n^3 +3757*n^2 -1268*n -1806)*Self(n-2) -6*(5*n^2-5*n-28)*Self(n-3) + (n-5)*(n-3)*(25*n^2-45*n-28)*Self(n-4))/((n-1)*(n+1)*(25*n^2-95*n+42)): n in [1..41]]; // G. C. Greubel, Nov 24 2022

Join[{1}, 2*Rest@CoefficientList[InverseSeries[Series[(-1 -6*n -8*n^2 + (1+ 2*n)^2*Sqrt[1+4*n])/(2*(n +4*n^2 +4*n^3)), {n, 0, 40}]], n]]
Length/@ Flatten/@ NestList[# /. k_Integer :> Range[-Abs[k+1], Abs[k-1], 2] &, {0}, 12]

def replace(L): return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]
def aList(upto, L=[0]): return [1] + [len((L := replace(L))) for _ in range(upto)]
print(aList(12))  # Peter Luschny, Nov 16 2024

@CachedFunction
def a(n): # a = A084078
    if (n<4): return (1,2,4,10)[n]
    else: return (6*(35*n^2 -55*n -76)*a(n-1) +(275*n^4-770*n^3-203*n^2+1736*n-912)*a(n-2) -6*(5*n^2+5*n-28)*a(n-3) +(n-4)*(n-2)*(25*n^2+5*n-48)*a(n-4))/(n*(n+2)*(25*n^2-45*n-28))
[a(n) for n in range(41)] # G. C. Greubel, Nov 24 2022

a(2*n-1) = A027307(n), n >= 1.
a(n) = 2*A084075(n-1), n >= 1.
a(n) = ( 6*(35*n^2 -55*n -76)*a(n-1) + (275*n^4 -770*n^3 -203*n^2 +1736*n -912)*a(n-2) - 6*(5*n^2 +5*n -28)*a(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*a(n-4) )/(n*(n+2)*(25*n^2 -45*n -28)), for n >= 4. - G. C. Greubel, Nov 24 2022
a(2*n) = A032349(n+1), n >= 0. - Alexander Burstein, Nov 19 2023

A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 40, 524, 7824, 126228, 2143544, 37750812, 683194912, 12628104740, 237388091208, 4524456276524, 87228274533040, 1698091537435444, 33332913873239640, 659038408936005692, 13112372856351746112, 262338658739430857796, 5274545338183090647656

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A006319, A032349, A371675. A371677, A371678.

Cf. A260332.

a(n, r=2, t=4, 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)^2 * (1 + A(x)^(1/2)) )^2.

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

A371678 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 56, 1068, 23504, 561972, 14183880, 371911132, 10031990560, 276589937892, 7759696110808, 220805824681740, 6357540660485616, 184876232243020564, 5422016433851400552, 160187931368799105468, 4763038761416835095616, 142426926824923660491716

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A006319, A032349, A371675. A371676, A371677.

Cf. A371700.

a(n, r=2, t=6, 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 * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A371700.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+2,n)/(6*n+k+2).

cp's OEIS Frontend

A102230 Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.

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A102231 Column 1 of triangle A102230 and equals the convolution of A032349 with A032349 shift right.

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A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

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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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A100324 Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right.

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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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A365847 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ).

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A084078 Length of list created by n substitutions k -> Range[-abs(k+1), abs(k-1), 2] starting with {0}.

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