A144485 -id:A144485 - cp's OEIS frontend

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).

1, 3, 2, 15, 20, 6, 84, 168, 105, 20, 495, 1320, 1260, 504, 70, 3003, 10010, 12870, 7920, 2310, 252, 18564, 74256, 120120, 100100, 45045, 10296, 924, 116280, 542640, 1058148, 1113840, 680680, 240240, 45045, 3432

Offset: 0

F. Chapoton, Oct 06 2022

Each line should be the f-vector of a cellular complex. The sequence seems to give the coefficients in a binomial basis of the integer-valued polynomials (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2n)!).

The precise expansion is (x+1)*(x+2)*...*(x+2*n)*(x+1)*(x+2)*...*(x+n)/(n!*(2*n)!) = Sum_{k = 0..n} (-1)^k*T(n,k)*binomial(x+3*n-k, 3*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)^2*(x+2)^2*(x+3)^2*(x+4)*(x+5)*(x+6)/(3!*6!) = 84*binomial(x+9, 9) - 168*binomial(x+8, 8) + 105*binomial(x+7, 7) - 20*binomial(x+6, 6). - Peter Bala, Jun 25 2023

			As a triangle of numbers, this starts with
  1;
  3, 2;
  15, 20, 6;
  84, 168, 105, 20;
  495, 1320, 1260, 504, 70.
Here is an example for n=1 as coefficients (up to sign) in the binomial basis of integer-valued polynomials:
(x+1)*(x+2)*(x+1)/2 = 3*binomial(x+3,3)-2*binomial(x+2,2).

Row sums A026000. Cf. A000984, A005809 (k=0), A144485 (k=1), A033282, A110608, A243660.

A357613 := proc(n,k)
    binomial(2*n,k)*binomial(3*n-k,2*n) ;
end proc:
seq(seq(A357613(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jul 06 2023

Table[Binomial[2n,k]Binomial[3n-k,2n],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Oct 11 2023 *)

def a(n):
    return [binomial(2 * n, k) * binomial(3 * n - k, 2 * n)
            for k in range(n + 1)]

T(n,k) = binomial(2*n, k) * binomial(3*n - k, 2*n) for 0 <= k <= n

A378777 a(n) = n^2 * binomial(3*n, n).

Original entry on oeis.org

0, 3, 60, 756, 7920, 75075, 668304, 5697720, 47070144, 379632825, 3004501500, 23417943120, 180241588800, 1372689900036, 10360604899680, 77595170756400, 577241321893632, 4268838966063525, 31404136939468020, 229951212925133700, 1676737802322198000, 12180171012442098435

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.

Cf. A005809, A090763, A144485.

a[n_] := n^2 * Binomial[3*n, n]; Array[a, 25, 0]

PARI
```
a(n) = n^2 * binomial(3*n, n);
```

a(n) = n^2 * A005809(n).
a(n) = 3 * n * A090763(n-1) = 3 * n * A144485(n) / 2.
Sum_{n>=1} 1/a(n) = 6 * arctan(sqrt(3)/(2*phi-1))^2 - log((phi^3+1)/(phi+1)^3)^2/2, where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.1)).

cp's OEIS Frontend

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).

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A378777 a(n) = n^2 * binomial(3*n, n).

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cp's OEIS Frontend

A357613 Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A378777 a(n) = n^2 * binomial(3*n, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A357613 Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).