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

1,2

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

Links

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

Formula

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

Example

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)).

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A001263, A034856, A000108, A131428.

Sequence in context: A209145 A333650 A214984 * A210235 A138177 A101559

Adjacent sequences: A118973 A118974 A118975 * A118977 A118978 A118979

Keyword

nonn,tabl

Author

Gary W. Adamson, May 07 2006

Extensions

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

Status

approved