A142470 Triangle T(n, k) = ( (k+2)/(2*binomial(k+2, 2)^2) )*binomial(n, k)^2*binomial(n+1, k)*binomial(n+2, k), read by rows.

1, 1, 1, 1, 8, 1, 1, 30, 30, 1, 1, 80, 300, 80, 1, 1, 175, 1750, 1750, 175, 1, 1, 336, 7350, 19600, 7350, 336, 1, 1, 588, 24696, 144060, 144060, 24696, 588, 1, 1, 960, 70560, 790272, 1728720, 790272, 70560, 960, 1, 1, 1485, 178200, 3492720, 14669424, 14669424, 3492720, 178200, 1485, 1

Offset

0,5

Comments

Row sums are 1, 2, 10, 62, 462, 3852, 34974, 338690, 3452306, 36683660, 403472368, ...

From Peter Bala, May 08 2012: (Start)

Define the action of the operator L on a sequence { a(i) }_{0<=i<=n} by L{ a(i) }_{0<=i<=n} = { a(i)^2 - a(i-1)*a(i+1) }_{0<=i<=n} with the conventions a(-1) = a(n+1) = 0. Extend the action of L to a lower triangular array T by letting L act on the rows of T. Then L acting on Pascal's triangle A007318 produces the triangle of Narayana numbers A001263 and L applied to A001263 produces the present triangle.

Since the Narayana polynomials are real-rooted it follows by a theorem of Branden that the row polynomials of this array are also real-rooted.

(End)

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Petter Brändén, Iterated sequences and the geometry of zeros, arXiv:0909.1927 [math.CO], 2009-2010; J. Reine Angew. Math. 658 (2011), 115-131.

Formula

Let f(n, k) = binomial(n, k)*Product_{j=1.2} ( j!*(n+j)!/((k+j)!*(n-k+j)!) ), then T(n, k) = 2^(k-n)*f(n, k)*Sum_{j=k..n} binomial(n, j)*binomial(j, k) = binomial(n, k)*f(n, k).

From Peter Bala, May 08 2012: (Start)

T(n, k) = C(n, k)^2 * Product {i=1..2} i!*(n+i)!/((k+i)!*(n-k+i)!) = C(n, k)*C(n+2, k)*C(n+2, k+1)*C(n+2, k+2)/(C(n+2, 1)*C(n+2, 2)).

T(n, k) = 2/((n+1)*(n+2)*(n+3))*C(n, k)*C(n+1, k)*C(n+2, k+2)*C(n+3, k+1) = C(n, k)*A056939(n, k).

(End)

T(n, k) = ( (k+2)/(2*binomial(k+2, 2)^2) )*binomial(n, k)^2*binomial(n+1, k)*binomial(n+2, k). - G. C. Greubel, Apr 03 2021

Example

The triangle begins as:
  1;
  1,    1;
  1,    8,      1;
  1,   30,     30,       1;
  1,   80,    300,      80,        1;
  1,  175,   1750,    1750,      175,        1;
  1,  336,   7350,   19600,     7350,      336,       1;
  1,  588,  24696,  144060,   144060,    24696,     588,      1;
  1,  960,  70560,  790272,  1728720,   790272,   70560,    960,    1;
  1, 1485, 178200, 3492720, 14669424, 14669424, 3492720, 178200, 1485, 1;

Mathematica

f[n_, k_]:= f[n, k]= Binomial[n, k]*Product[j!*(n+j)!/((k+j)!*(n-k+j)!), {j, 1, 2}];
T[n_, k_]:= Binomial[n, k]*f[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 03 2021 *)

Prog

(Magma)
A142470:= func< n, k | ( (k+2)/(2*Binomial(k+2, 2)^2) )*Binomial(n, k)^2*Binomial(n+1, k)*Binomial(n+2, k) >;
[A142470(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 03 2021
(Sage)
def A142470(n, k): return (2/((k+1)^2*(k+2)))*Binomial(n, k)^2*Binomial(n+1, k)*Binomial(n+2, k)
flatten([[A142470(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 03 2021

Crossrefs

Cf. A001263, A008459, A056939.

Sequence in context: A181543 A141696 A178122 * A168523 A144439 A157208

Adjacent sequences: A142467 A142468 A142469 * A142471 A142472 A142473

Keyword

nonn,tabl

Author

Roger L. Bagula, Sep 20 2008

Extensions

Edited by G. C. Greubel, Apr 03 2021

Status

approved