A363419 Square array read by ascending antidiagonals: T(n,k) = 1/n * [x^k] 1/((1 - x)*(1 - x^2))^(n*k) for n, k >= 1.

1, 1, 5, 1, 7, 19, 1, 9, 46, 85, 1, 11, 82, 327, 376, 1, 13, 127, 793, 2376, 1715, 1, 15, 181, 1547, 7876, 17602, 7890, 1, 17, 244, 2653, 19376, 79686, 132056, 36693, 1, 19, 316, 4175, 40001, 247205, 816684, 1000263, 171820, 1, 21, 397, 6177, 73501, 614389, 3195046, 8450585, 7632433, 809380

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.

Links

Table of n, a(n) for n=0..54.

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

Row 1 A348410. Cf. A362724 - A362733, A363418.

Sequence in context: A320905 A193860 A211849 * A222182 A126155 A021197

Adjacent sequences: A363416 A363417 A363418 * A363420 A363421 A363422

Keyword

nonn,tabl,easy

Author

Peter Bala, Jun 13 2023

Status

approved