A361027 Table of generalized de Bruijn's numbers (A006480) read by ascending antidiagonals.

2, 30, 3, 560, 20, 20, 11550, 210, 75, 210, 252252, 2772, 504, 504, 2772, 5717712, 42042, 4620, 2352, 4620, 42042, 133024320, 700128, 51480, 15840, 15840, 51480, 700128, 3155170590, 12471030, 656370, 135135, 81675, 135135, 656370, 12471030, 75957810500, 233716340, 9237800

Offset

0,1

Comments

The Catalan numbers A000108 are given by the formula Catalan(n) = (2*n)!/(n!*(n + 1)!). Gessel (1992) considered generalized Catalan numbers defined by Catalan(r,n) = J(r) * (2*n)!/(n!*(n + r + 1)!), where J(r) = (2*r + 2)!/(2*(r + 1)!) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. Gessel's generalized Catalan numbers are particular cases of super ballot numbers. See A135573 for a table of these generalized Catalan numbers.

For this table we carry out an analogous construction using the de Bruijn numbers B(n) = (3*n)!/n!^3 = A006480(n) in place of the central binomial numbers. We define the generalized de Bruijn number B(r,n), r = 0, 1, 2, ..., by B(r,n) = F(r) * (3*n)!/(n!*(n + r + 1)!^2), where choosing F(r) = (3*r + 3)!/(3*(r + 1)!) = (3^r)*Product_{j = 0..r} (3*j + 1)*(3*j + 2) appears to produce integer values for these quantities. We have verified this for rows 0, 1, 2 and 3 of the table.

An alternative expression for the generalized de Bruijn numbers is B(r,n) = G(r,n) * B(n+r+1), where G(r) = (1/3)*Product_{j = 0..r} ( (3*j + 1)*(3*j + 2)/((3*n + 3*j + 1)*(3*n + 3*j + 2)) ).

The rows of the square array below are the sequences of generalized de Bruijn numbers {B(0,k)}, {B(1,k)}, {B(2,k)}, ....

References

N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Links

Table of n, a(n) for n=0..38.

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.

Formula

T(n,k) = (3*n + 3)!/(3*(n + 1)!) * (3*k)!/(k!*(k + n + 1)!^2), n, k >= 0.

T(n,k) = (1/3)*27^(n+1+k)*binomial(n+1/3, n+1+k)*binomial(n+2/3, n+1+k).

T(n,k) = (1/(2*Pi))^2 * 1/27^(n+k+1) * Integral_{x = 0..27} (27 - x)^(n+2/3)*x^(k-2/3) dx * Integral_{x = 0..27} (27 - x)^(n+1/3)*x^(k-1/3) dx.

P-recursive: (n + k + 1)^2*T(n,k) = 3*(3*k - 1)*(3*k - 2)*T(n,k-1) with T(n,0) = 1/(n+1)!^2 * (3*n + 3)!/(3*(n + 1)!).

(n + k + 1)^2*T(n,k) = 3*(3*n + 1)*(3*n + 2)*T(n-1,k) with T(0,k) = 2*(k + 1)*(3*k)!/(k + 1)!^3.

T(n,0) = A208881(n+1).

Example

The square array with rows n >= 0 and columns k >= 0 begins:
  n\k|       0       1       2        3        4         5         6 ...
  ----------------------------------------------------------------------
   0 |       2       3      20      210     2772     42042    700128 ...
   1 |      30      20      75      504     4620     51480    656370 ...
   2 |     560     210     504     2352    15840    135135   1361360 ...
   3 |   11550    2772    4620    15840    81675    550550   4492488 ...
   4 |  252252   42042   51480   135135   550550   3006003  20271888 ...
   5 | 5717712  700128  656370  1361360  4492488  20271888  ...
  ...
As a triangle:
 Row
  0 |        2
  1 |       30       3
  2 |      560      20     20
  3 |    11550     210     75    210
  4 |   252252    2772    504    504   2772
  5 |  5717712   42042   4620   2352   4620   42042
  ...

Maple

# as a square array
T := proc (n, k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
n+k+1); end proc:
for n from 0 to 10 do seq(T(n, k), k = 0..10); end do;
# as a triangle
T := proc (n, k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
n+k+1); end proc:
for n from 0 to 10 do seq(T(n-k, k), k = 0..n); end do;

Crossrefs

A208881 (column 1), A361028(row 0), A361029(row 1), A361030(row 2), A361031(row 3).

Cf. A006480, A024488, A135573.

Sequence in context: A331427 A058988 A292879 * A267131 A078690 A228937

Adjacent sequences: A361024 A361025 A361026 * A361028 A361029 A361030

Keyword

nonn,tabl,easy

Author

Peter Bala, Feb 28 2023

Status

approved