OFFSET
0,3
COMMENTS
The n-th row sequence {T(n, k) : k >= 1} satisfies the Gauss congruences, that is, T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^r ) for all primes p and positive integers m and r.
We conjecture that each row sequence satisfies the stronger supercongruences T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^(3*r) ) for all primes p >= 5 and positive integers m and r.
The table can be extended to negative values of n, and the row sequences also appear to satisfy the above supercongruences.
REFERENCES
R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
FORMULA
T(n,k) = (1/n)*Sum_{j = 0..floor(k/2)} binomial(n*k + j - 1, j)*binomial((n+1)*k - 2*j - 1, k - 2*j).
Define E(n,x) = exp( Sum_{j >= 1} T(n,j)*x^j/j ). Then T(n+1,k) = [x^k] E(n,x)^k.
E(n,x) = (1/x) * the series reversion of x/E(n-1,x) for n >= 2.
E(n,x)^n = (1/x) * the series reversion of x*((1 - x)(1 - x^2))^n.
T(n,k) = (1/n)*binomial(n*k+k-1,k) * hypergeom([n*k, -k/2, (1 - k)/2], [(1 - (n+1)*k)/2, (2 - (n+1)*k)/2], 1) except when n = 1 and k = 1 or 2.
The o.g.f. for row n is the diagonal of the bivariate rational function (1/n) * t*f(x)^n/(1 - t*f(x)^n), where f(x) = 1/((1 - x)*(1 - x^2)), and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
EXAMPLE
The square array begins
n\k | 1 2 3 4 5 6 7
- - + - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 | 1 5 19 85 376 1715 7890 ... (A348410)
2 | 1 7 46 327 2376 17602 132056 ...
3 | 1 9 82 793 7876 79686 816684 ...
4 | 1 11 127 1547 19376 247205 3195046 ...
5 | 1 13 181 2653 40001 614389 9560097 ...
6 | 1 15 244 4175 73501 1318236 23952720 ...
7 | 1 17 316 6177 124251 2546288 52867620 ...
8 | 1 19 397 8723 197251 4544407 106076867 ...
9 | 1 21 487 11877 298126 7624551 197571088 ...
10 | 1 23 586 15703 433126 12172550 346618308 ...
Array extended to negative values of n:
n\k | 1 2 3 4 5 6 7
- - + - - - - - - - - - - - - - - - - - - - - - - - - - - -
-5 | 1 -7 46 -247 626 8642 -194480 ...
-4 | 1 -5 19 -5 -874 11569 -105300 ...
-3 | 1 -3 1 77 -749 4641 -19893 ...
-2 | 1 -1 -8 63 -249 440 1716 ...
-1 | 1 1 -8 17 1 -116 344 ... (-A234839)
MAPLE
# display as a square array
T := (n, k) -> (1/n)*add( (-1)^(k+j) * binomial(-n*k, j)*binomial(-n*k, k-2*j) , j = 0..floor(k/2)): for n from 1 to 10 do seq(T(n, k), k = 1..10) end do;
# display as a sequence
seq(seq(T(n+1-i, i), i = 1..n), n = 1..10);
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Jun 13 2023
STATUS
approved