The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #17 Mar 13 2023 07:20:50
%S 2,30,3,560,20,20,11550,210,75,210,252252,2772,504,504,2772,5717712,
%T 42042,4620,2352,4620,42042,133024320,700128,51480,15840,15840,51480,
%U 700128,3155170590,12471030,656370,135135,81675,135135,656370,12471030,75957810500,233716340,9237800
%N Table of generalized de Bruijn's numbers (A006480) read by ascending antidiagonals.
%C 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.
%C 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.
%C 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)) ).
%C The rows of the square array below are the sequences of generalized de Bruijn numbers {B(0,k)}, {B(1,k)}, {B(2,k)}, ....
%D N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.
%H Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>, J. Symbolic Comp., 14 (1992), 179-194.
%F T(n,k) = (3*n + 3)!/(3*(n + 1)!) * (3*k)!/(k!*(k + n + 1)!^2), n, k >= 0.
%F T(n,k) = (1/3)*27^(n+1+k)*binomial(n+1/3, n+1+k)*binomial(n+2/3, n+1+k).
%F 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.
%F 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)!).
%F (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.
%F T(n,0) = A208881(n+1).
%e The square array with rows n >= 0 and columns k >= 0 begins:
%e n\k| 0 1 2 3 4 5 6 ...
%e ----------------------------------------------------------------------
%e 0 | 2 3 20 210 2772 42042 700128 ...
%e 1 | 30 20 75 504 4620 51480 656370 ...
%e 2 | 560 210 504 2352 15840 135135 1361360 ...
%e 3 | 11550 2772 4620 15840 81675 550550 4492488 ...
%e 4 | 252252 42042 51480 135135 550550 3006003 20271888 ...
%e 5 | 5717712 700128 656370 1361360 4492488 20271888 ...
%e ...
%e As a triangle:
%e Row
%e 0 | 2
%e 1 | 30 3
%e 2 | 560 20 20
%e 3 | 11550 210 75 210
%e 4 | 252252 2772 504 504 2772
%e 5 | 5717712 42042 4620 2352 4620 42042
%e ...
%p # as a square array
%p T := proc (n,k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
%p n+k+1); end proc:
%p for n from 0 to 10 do seq(T(n,k), k = 0..10); end do;
%p # as a triangle
%p T := proc (n,k) (1/3)*27^(n+k+1)*binomial(n+1/3, n+k+1)*binomial(n+2/3,
%p n+k+1); end proc:
%p for n from 0 to 10 do seq(T(n-k,k), k = 0..n); end do;
%Y A208881 (column 1), A361028(row 0), A361029(row 1), A361030(row 2), A361031(row 3).
%Y Cf. A006480, A024488, A135573.
%K nonn,tabl,easy
%O 0,1
%A _Peter Bala_, Feb 28 2023