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A363419
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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.
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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)
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MAPLE
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# 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);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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