A062344 Triangle of binomial(2*n, k) with n >= k.

1, 1, 2, 1, 4, 6, 1, 6, 15, 20, 1, 8, 28, 56, 70, 1, 10, 45, 120, 210, 252, 1, 12, 66, 220, 495, 792, 924, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620

Offset

0,3

Comments

From Wolfdieter Lang, Sep 19 2012: (Start)

The triangle a(n,k) appears in the formula F(2*l+1)^(2*n) = (sum(a(n,k)*L(2*(n-k)*(2*l+1)),k=0..n-1) + a(n,n))/5^n, n>=0, l>=0, with F=A000045 (Fibonacci) and L=A000032 (Lucas).

The signed triangle as(n,k):=a(n,k)*(-1)^k appears in the formula F(2*l)^(2*n) = (sum(as(n,k)*L(4*(n-k)*l),k=0..n-1) + as(n,n))/5^n, n>=0, l>=0. Proof with the Binet-de Moivre formula for F and L and the binomial formula. (End)

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.

C. Lanczos, Applied Analysis (Annotated scans of selected pages)

Index entries for triangles and arrays related to Pascal's triangle

Formula

a(n,k) = a(n,k-1)*((2n+1)/k-1) with a(n,0)=1.

G.f.: 1/((1-sqrt(1-4*x*y))^4/(16*x*y^2) + sqrt(1-4*x*y) - x). - Vladimir Kruchinin, Jan 26 2021

Example

Rows start
  (1),
  (1,2),
  (1,4,6),
  (1,6,15,20)
  etc.
Row n=2, (1,4,6):
F(2*l+1)^4 = (1*L(4*(2*l+1)) + 4*L(2*(2*l+1)) + 6)/25,
F(2*l)^4 = (1*L(8*l) - 4*L(4*l) + 6)/25, l>=0, F=A000045, L=A000032. See a comment above. - Wolfdieter Lang, Sep 19 2012

Mathematica

Flatten[Table[Binomial[2 n, k], {n, 0, 20}, {k, 0, n}]] (* G. C. Greubel, Jun 28 2018 *)

Prog

(Maxima) create_list(binomial(2*n, k), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */
(PARI) for(n=0, 20, for(k=0, n, print1(binomial(2*n, k), ", "))) \\ G. C. Greubel, Jun 28 2018
(Magma) [[Binomial(2*n, k): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Jun 28 2018

Crossrefs

Columns include (sometimes truncated) A000012, A005843, A000384, A002492, A053134 etc. Right hand side includes A000984, A001791, A002694, A002696 etc. Row sums are A032443. Row alternate differences (e.g., 6-4+1=3 or 20-15+6-1=10) are A001700.

Cf. A122366.

a(2n,n) gives A005810.

Sequence in context: A033884 A208915 A199704 * A208759 A033877 A059369

Adjacent sequences: A062341 A062342 A062343 * A062345 A062346 A062347

Keyword

nonn,tabl

Author

Henry Bottomley, Jul 06 2001

Status

approved