A131428 -id:A131428 - cp's OEIS frontend

A131427 A000108(n) preceded by n zeros.

1, 0, 1, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 0, 429, 0, 0, 0, 0, 0, 0, 0, 0, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 58786, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 208012

Offset: 0

Gary W. Adamson, Jul 10 2007

Triangle given by A000004 DELTA A000012 where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 12 2007

T(n,k) is the number of Dyck paths of semilength n having exactly k U=(1,1) steps. - Alois P. Heinz, Jun 09 2014

			First few rows of the triangle are:
1;
0, 1;
0, 0, 2;
0, 0, 0, 5;
0, 0, 0, 0, 14;
0, 0, 0, 0, 0, 42;
...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A131428, A131429, A000108, A243752.

T:= (n, k)-> `if`(kAlois P. Heinz, Jun 09 2014

T[n_, n_] := CatalanNumber[n]; T[, ] = 0;
Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 20 2016 *)

A000108(n) preceded by n zeros, as an infinite lower triangular matrix.

More terms from Philippe Deléham, Oct 16 2008

A131429 Triangle read by rows: T(n,k) = C(n) + C(k) - 1 where C(n) = A000108(n) are the Catalan numbers, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 5, 6, 9, 14, 14, 15, 18, 27, 42, 42, 43, 46, 55, 83, 132, 132, 133, 136, 145, 173, 263, 429, 429, 430, 433, 442, 470, 560, 857, 1430, 1430, 1431, 1434, 1443, 1471, 1561, 1858, 2859, 4862, 4862, 4863, 4866, 4875, 4903, 4993, 5290, 6291, 9723

Offset: 0

Gary W. Adamson, Jul 10 2007

Left column = Catalan numbers, A000108. Right border = 2*A000108 - 1. Row sums = A131430: (1, 2, 7, 25, 88, 311, 1114, ...).

			First few rows of the triangle are:
   1;
   1,  1;
   2,  2,  3;
   5,  5,  6,  9;
  14, 14, 15, 18, 27;
  42, 42, 43, 46, 55, 83;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (first 50 rows)

Column k=0 is A000108.
Row sums are A131430.
Cf. A131427, A131428.

T(n,k)=if(k<=n, binomial(2*n,n)/(n+1) + binomial(2*k,k)/(k+1) - 1, 0) \\ Andrew Howroyd, Sep 01 2018

Equals (A000012 * A131427) + (A131427 * A000012) - A000012 as infinite lower triangular matrices.

Name clarified by Andrew Howroyd, Sep 01 2018

A381846 Area of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 0, 6, 180, 4914, 142926, 4547796, 157355484, 5842280730, 229795151586, 9475645552620, 406294220860710, 18000809380947036, 820011973477512900, 38258534425043501640, 1822437060664227775020, 88405827105467677196970, 4358079981772447955690490, 217935769988152202470568700

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 06 2025

			For n=3, the short leg is A383615(3,1) = 3 and the long leg is A383615(3,2) = 4  so the area is then a(4) = (3 * 4)/2 = 6.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000108, A383615, A383616, A131428.

a=Table[(2n)!/(n!(n+1)!),{n,0,18}];Apply[Join,Map[{#(#-1)(2#-1)}&,a]]

a(n) = (A383615(n,1) * A383615(n,2))/2.

a(n) = (2n)!/(n!(n+1)!)*((2n)!/(n!(n+1)!) - 1)*(2*(2n)!/(n!(n+1)!) - 1).

A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.

Original entry on oeis.org

1, 1, 3, 129, 1587, 39443, 1125383, 30211457, 1107074979, 36214609683, 1433494688871, 54495716261011, 2275005440977063, 95146470595975399, 4170974287982618639, 185640304224109725569, 8492643748223480148419, 395051289603660979274339, 18726850582009755291702599

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Cf. A131428, A382434.

Cf. A080233, A156644, A382433.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^6, j=0..n/2)-1:
seq(a(n), n=0..18);  # Alois P. Heinz, Mar 25 2025

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^6);

from math import comb
def A382435(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..n} A080233(n,k)^6 = Sum_{k=0..n} A156644(n,k)^6.

a(n) = 2 * A382433(n) - 1.

A383615 Length of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

Original entry on oeis.org

4, 40, 364, 3444, 34584, 367224, 4086940, 47268364, 564177640, 6911470020, 86537568264, 1103799334200, 14305253278320, 187980019758360, 2500329584942460, 33615542888998620, 456277454520102600, 6246438361923425820, 86175353763393711960, 1197196443738946826760

Offset: 2

Miguel-Ángel Pérez García-Ortega, May 02 2025

			Triangles begin:
  n=2:      3,     4,     5;
  n=3:      9,    40,    41.
This sequence is column 2.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000108, A131428 (short leg), A383616 (semiperimeter), A381846 (area).

a(n) = 2*C(n)*(C(n) - 1) where C(n) = A000108(n).

A383616 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 1, 6, 45, 378, 3486, 34716, 367653, 4088370, 47273226, 564194436, 6911528806, 86537776276, 1103800077100, 14305255952760, 187980029453205, 2500329620300130, 33615543018643410, 456277454997741300, 6246438363690689010, 86175353769957832380, 1197196443763413093780, 16738118900201817535560

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 02 2025

			For n=3, the short leg is A383615(3,1) = 3, the long leg is A383615(3,2) = 4 and the hypotenuse is A383615(3,3) = 5 so the semiperimeter is then a(3) = (3 + 4 + 5)/2 = 6.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000108, A383615, A381846, A131428.

a=Table[(2n)!/(n!(n+1)!),{n,0,22}];Apply[Join,Map[{#(2#-1)}&,a]]

a(n) = (A383615(n,1) + A383615(n,2) + A383615(n,3)) / 2.

a(n) = ((2n)! / (n!*(n+1)!)) * (2*(2n)! / (n!*(n+1)!) - 1).

A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

Original entry on oeis.org

1, 1, 4, 17, 86, 472, 2752, 16753, 105394, 680366, 4484360, 30067160, 204508240, 1408057120, 9796738304, 68786005361, 486845236106, 3470187822754, 24891491746792, 179556655434382, 1301857088258836, 9482632068303296, 69361538748381824, 509303099950899352

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A129123, A382433, A382443, A382446.
Cf. A131428, A178824.
Cf. A000172, A086618, A238115.

[ &+[Binomial(n, k)^2 * (Binomial(n, k) - (k gt 0 select Binomial(n, k-1) else 0)) : k in [0..n]] : n in [0..20] ]; // Vincenzo Librandi, Mar 27 2025

Table[Sum[Binomial[n,k]^2*(Binomial[n,k]-Binomial[n,k-1]),{k,0,n}],{n,0,20}] (* Vincenzo Librandi, Mar 27 2025 *)

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

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

a(n) ~ 2^(3*n+3) / (Pi * 3^(3/2) * n^2). - Vaclav Kotesovec, Mar 26 2025

A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

Original entry on oeis.org

1, 1, 3, 33, 195, 1763, 15623, 156257, 1630947, 17911299, 203739015, 2389928995, 28749060871, 353362388551, 4424242664975, 56290517376737, 726355164976547, 9490129871680355, 125375330053632455, 1672895457018337859, 22522481793315373319, 305695116823973096519

Offset: 0

Seiichi Manyama, Mar 25 2025

nonn

Cf. A131428, A382435.

Cf. A080233, A129123, A156644.

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> 2*add(b(n, n-2*j)^4, j=0..n/2)-1:
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025

a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^4);

from math import comb
def A382434(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1))<<1)-1 # Chai Wah Wu, Mar 25 2025

a(n) = Sum_{k=0..n} A080233(n,k)^4 = Sum_{k=0..n} A156644(n,k)^4.
a(n) = 2 * A129123(n) - 1.
D-finite with recurrence n*(n+1)^3*a(n) -2*n*(11*n^3-17*n^2+5*n+5)*a(n-1) -4*(n-1)*(70*n^3-365*n^2+527*n-162)*a(n-2) +8*(n-2)*(584*n^3-5020*n^2+14111*n-13059)*a(n-3) +1344*(4*n-11)*(4*n-13)*(-3+n)^2*a(n-4) +9*(2875*n^4-33975*n^3+149945*n^2-293541*n+215336)=0. - R. J. Mathar, Mar 31 2025

A118976 Triangle read by rows: T(n,k) = binomial(n-1,k-1)binomial(n,k-1)/k + binomial(n-1,k)binomial(n,k)/(k+1) (1 <= k <= n). In other words, to each entry of the Narayana triangle (A001263) add the entry on its right.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 7, 12, 7, 1, 11, 30, 30, 11, 1, 16, 65, 100, 65, 16, 1, 22, 126, 280, 280, 126, 22, 1, 29, 224, 686, 980, 686, 224, 29, 1, 37, 372, 1512, 2940, 2940, 1512, 372, 37, 1, 46, 585, 3060, 7812, 10584, 7812, 3060, 585, 46, 1, 56, 880, 5775, 18810, 33264, 33264, 18810, 5775, 880, 56, 1

Offset: 1

Gary W. Adamson, May 07 2006

Sum of entries in row n = 2*Cat(n)-1, where Cat(n) are the Catalan numbers (A000108).

Row sums = A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,   1;
   7, 12,   7,  1;
  11, 30,  30, 11,  1;
  16, 65, 100, 65, 16, 1;
...
Row 4 of the triangle = (7, 12, 7, 1), derived from row 4 of the Narayana triangle, (1, 6, 6, 1): = ((1+6), (6+6), (6+1), (1)).

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001263, A034856, A000108, A131428.

B:=Binomial; Flat(List([1..12], n-> List([1..n], k-> B(n-1,k-1)*B(n,k-1)/k + B(n-1,k)*B(n,k)/(k+1) ))); # G. C. Greubel, Aug 12 2019

B:=Binomial; [B(n-1,k-1)*B(n,k-1)/k + B(n-1,k)*B(n,k)/(k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 12 2019

T:=(n,k)->binomial(n-1,k-1)*binomial(n,k-1)/k+binomial(n-1,k) *binomial(n,k)/ (k+1): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
# Alternatively:
gf := 1 - ((1/2)*(x + 1)*(sqrt((x*y + y - 1)^2 - 4*y^2*x) + x*y + y - 1))/(y*x):
sery := series(gf, y, 10): coeffy := n -> expand(coeff(sery, y, n)):
seq(print(seq(coeff(coeffy(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Oct 21 2020

With[{B=Binomial}, Table[B[n-1,k-1]*B[n,k-1]/k + B[n-1,k]*B[n,k]/(k+1), {n,12}, {k,n}]//Flatten] (* G. C. Greubel, Aug 12 2019 *)

T(n,k) = b=binomial; b(n-1,k-1)*b(n,k-1)/k + b(n-1,k)*b(n,k)/(k+1);
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 12 2019

def T(n, k):
    b=binomial
    return b(n-1,k-1)*b(n,k-1)/k + b(n-1,k)*b(n,k)/(k+1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Aug 12 2019

G.f.: A001263(x, y)*(x + x*y) + x*y. - Vladimir Kruchinin, Oct 21 2020

Edited by N. J. A. Sloane, Nov 29 2006

A131430 Row sums of triangle A131429.

Original entry on oeis.org

1, 2, 7, 25, 88, 311, 1114, 4050, 14917, 55528, 208459, 787920, 2994655, 11433998, 43824437, 168520201, 649840740, 2512011359, 9731179138, 37768570827, 146833956266, 571711568905, 2229035221824, 8701426599353, 34005503702176, 133030452858279, 520905732440782

Offset: 0

Gary W. Adamson, Jul 10 2007

nonn

			a(3) = 25 = sum of row 3 terms of triangle A131429: (5 + 5 + 6 + 9).

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000108, A131427, A131428, A131429.

a(n)={binomial(2*n,n) - n - 1 + sum(k=0, n, binomial(2*k,k)/(k+1))} \\ Andrew Howroyd, Aug 28 2018

a(n) = binomial(2*n, n) - (n+1) + Sum_{k=0..n} binomial(2*k, k)/(k+1). - Andrew Howroyd, Aug 28 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 28 2018

cp's OEIS Frontend

A131427 A000108(n) preceded by n zeros.

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A131429 Triangle read by rows: T(n,k) = C(n) + C(k) - 1 where C(n) = A000108(n) are the Catalan numbers, 0 <= k <= n.

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A381846 Area of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

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A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.

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A383615 Length of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 = A000108(n) and its long leg and hypotenuse are consecutive natural numbers.

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A383616 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

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A382434 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^4.

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A118976 Triangle read by rows: T(n,k) = binomial(n-1,k-1)binomial(n,k-1)/k + binomial(n-1,k)binomial(n,k)/(k+1) (1 <= k <= n). In other words, to each entry of the Narayana triangle (A001263) add the entry on its right.