A085374 -id:A085374 - cp's OEIS frontend

A085375 a(n) = binomial(2n+1, n+1)binomial(n+4, 4).

1, 15, 150, 1225, 8820, 58212, 360360, 2123550, 12033450, 66050270, 353068716, 1845586470, 9464546000, 47738754000, 237329805600, 1164893795820, 5653161067950, 27157342385250, 129275302348500, 610315506350550, 2859764086899720, 13308425945529000

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 26 2003

Chai Wah Wu, Table of n, a(n) for n = 0..500

Cf. A001700, A002457, A085373, A085374.

Cf. A000332, A000984.

seq(binomial(2*n+1, n+1)*binomial(n+4, 4), n=0..20); # Zerinvary Lajos, Jan 18 2007

Table[Binomial[2*n + 1, n + 1] * Binomial[n + 4, 4], {n, 0, 30}]

from _future_ import division
A085375_list, b = [], 1
for n in range(501):
    A085375_list.append(b)
    b = b*2*(n+5)*(2*n+3)//((n+1)*(n+2)) # Chai Wah Wu, Jan 26 2016

a(n+1) = a(n)*2*(n+5)*(2*n+3)/((n+1)*(n+2)). - Chai Wah Wu, Jan 26 2016

G.f.: (1 - 3*x + 6*x^2 - 5*x^3) / (1 - 4*x)^(9/2). - Ilya Gutkovskiy, Nov 17 2021

A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 14, 30, 12, 1, 42, 140, 100, 20, 1, 132, 630, 700, 250, 30, 1, 429, 2772, 4410, 2450, 525, 42, 1, 1430, 12012, 25872, 20580, 6860, 980, 56, 1, 4862, 51480, 144144, 155232, 74088, 16464, 1680, 72, 1, 16796, 218790, 772200, 1081080, 698544

Offset: 1

Gary W. Adamson, Jun 07 2004

The first three columns are A000108 (the Catalan numbers), A002457 and A085374.

			The first 3 rows are 1; 2, 1; 5, 6, 1; since the first 3 rows of the Narayana triangle in matrix format are M = [1 0 0 / 1 1 0 / 1 3 1]. Then M^2 = [1 0 0 / 2 1 0 / 5 6 1].
Triangle starts:
   1;
   2,   1;
   5,   6,   1;
  14,  30,  12,  1;
  42, 140, 100, 20, 1;
  ...

Cf. A000108, A001263, A002457, A085374.

t[n_, k_] = Sum[1/(i*k)*(Binomial[i-1, k-1]*Binomial[i, k-1]* Binomial[n-1, i-1]*Binomial[n, i-1]), {i, k, n}];
Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]][[1;;50]] (* Jean-François Alcover, Jul 21 2011 *)

T(n, k) = Sum_{i = k..n} A001263(n, i)*A001263(i, k).

T(n, n-1) = n*(n-1).

Edited and extended by David Wasserman, Sep 24 2004

cp's OEIS Frontend

A085375 a(n) = binomial(2n+1, n+1)binomial(n+4, 4).

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A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

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cp's OEIS Frontend

A085375 a(n) = binomial(2*n+1, n+1)*binomial(n+4, 4).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A085375 a(n) = binomial(2n+1, n+1)binomial(n+4, 4).