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.

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

Author

Gary W. Adamson, May 07 2006

Keywords

• nonn
• tabl

Comments

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

Examples

			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)).
		

Links

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

Crossrefs

Cf. A001263, A034856, A000108, A131428.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Sage

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

Formula

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

Extensions

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