A181543 Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.

1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 216, 64, 1, 1, 125, 1000, 1000, 125, 1, 1, 216, 3375, 8000, 3375, 216, 1, 1, 343, 9261, 42875, 42875, 9261, 343, 1, 1, 512, 21952, 175616, 343000, 175616, 21952, 512, 1, 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1

Offset

0,5

Comments

Diagonal of rational function R(x,y,z,t) = 1/(1 + y + z + x*y + y*z + t*x*z + (t+1)*x*y*z) with respect to x, y, z, i.e., T(n,k) = [(xyz)^n*t^k] R(x,y,z,t). - Gheorghe Coserea, Jul 01 2018

Links

Indranil Ghosh, Rows 0..120 of triangle, flattened

Jeffrey S. Geronimo, Hugo J. Woerdeman, and Chung Y. Wong, The autoregressive filter problem for multivariable degree one symmetric polynomials, arXiv:2101.00525 [math.CA], 2021.

Formula

Row sums equal A000172, the Franel numbers.

Central terms are A002897(n) = C(2n,n)^3.

Antidiagonal sums equal A181545;

The g.f. of the antidiagonal sums is Sum_{n>=0} (3n)!/(n!)^3 * x^(3n)/(1-x-x^2)^(3n+1).

G.f. for column k: [Sum_{j=0..2k} A181544(k,j)*x^j]/(1-x)^(3k+1), where the row sums of A181544 equals De Bruijn's s(3,n) = (3n)!/(n!)^3.

G.f.: A(x,y) = Sum_{n>=0} (3n)!/n!^3 * x^(2n)*y^n/(1-x-x*y)^(3n+1). - Paul D. Hanna, Nov 04 2010

Example

Triangle begins:
  1;
  1,   1;
  1,   8,     1;
  1,  27,    27,      1;
  1,  64,   216,     64,       1;
  1, 125,  1000,   1000,     125,       1;
  1, 216,  3375,   8000,    3375,     216,      1;
  1, 343,  9261,  42875,   42875,    9261,    343,     1;
  1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;
  1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;
  ...

Maple

T:= (n, k)-> binomial(n, k)^3:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 06 2021

Mathematica

Flatten[Table[Binomial[n, k]^3, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, May 23 2011 *)

Prog

(PARI) T(n, k)=binomial(n, k)^3
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print())
(PARI) T(n, k)=polcoeff(polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(3*m+1)), n, x), k, y)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Nov 04 2010
(PARI)
diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));
  return(a);
};
x='x; y='y; z='z; t='t;
concat(apply(Vec, diag(1/(1 + y + z + x*y + y*z + t*x*z + (t+1)*x*y*z), 10, [x, y, z]))) \\ Gheorghe Coserea, Jul 01 2018

Crossrefs

Cf. A000172 (row sums), A181545 (antidiagonal sums), A002897, A181544, A248658.

Variants: A008459, A007318.

Sequence in context: A220718 A176283 A323324 * A141696 A178122 A142470

Adjacent sequences: A181540 A181541 A181542 * A181544 A181545 A181546

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 30 2010

Status

approved