OFFSET
0,6
COMMENTS
T(n,k) is the number of basis elements in the order-n Brauer algebra that have propagation number k. - John M. Campbell, Dec 08 2021
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
T(n,k) = k!*binomial(n,k)^2*(n-k-1)!!^2 if n-k is even; T(n,k) = 0 if n-k is odd.
Sum of row n = (2n-1)!! = A001147(n).
T(n,n) = n! = A000142(n).
T(2n,0) = A001818(n).
Sum_{k>=0} k*T(n,k) = n^2*(2n-3)!! = A161120(n).
From Natalia L. Skirrow, Mar 18 2026: (Start)
E.g.f.: 1/sqrt((1-x*y)^2 - x^2).
D-finite with T(n,k) = (n-1)*T(n-2,k) + (k*(k+1) + n*(3*(n-k)-2)) * T(n-1,k-1)/(n-1) for n > 0.
Diagonals are D-finite with T(n,k) = n^2*T(n-1,k-1)/k for k > 0.
(End)
EXAMPLE
T(3,1)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
n\k| 0 1 2 3 4 5 6 7
---+-------------------------------------
0 | 1
1 | 0 1
2 | 1 0 2
3 | 0 9 0 6
4 | 3 0 72 0 24
5 | 0 75 0 600 0 120
6 |15 0 1350 0 5400 0 720
7 | 0 735 0 22050 0 52920 0 5040
MAPLE
T := proc (n, k) if `mod`(n-k, 2) = 1 then 0 else binomial(n, k)^2*factorial(k)*(product(2*j-1, j = 1 .. (1/2)*n-(1/2)*k))^2 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
MATHEMATICA
CoefficientList[CoefficientList[Normal[Series[1/Sqrt[(1-x*y)^2-x^2], {x, 0, 8}]], x]*Range[0, 8]!, y]//MatrixForm (* Natalia L. Skirrow, Mar 18 2026 *)
PROG
(PARI) dfo(n) = if (n<0, (-1)^n/dfo(-n), (2*n)! / n! / 2^n); \\ A001147
T(n, k) = if ((n-k)%2, 0, k!*binomial(n, k)^2*dfo((n-k)/2)^2);
row(n) = vector(n+1, k, T(n, k-1)) \\ Michel Marcus, Dec 09 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jun 02 2009
STATUS
approved