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A316674 Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 13, 26, 4, 1, 1, 75, 818, 252, 8, 1, 1, 541, 47834, 64324, 2568, 16, 1, 1, 4683, 4488722, 42725052, 5592968, 26928, 32, 1, 1, 47293, 617364026, 58555826884, 44418808968, 515092048, 287648, 64, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n. - Andrew Howroyd, Jan 23 2020
LINKS
FORMULA
A(n,k) = A262809(n,k) * A011782(n) for k>0, A(n,0) = 1.
A(n,k) = Sum_{j=0..n*k} binomial(j+n-1,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 13, 75, 541, ...
1, 2, 26, 818, 47834, 4488722, ...
1, 4, 252, 64324, 42725052, 58555826884, ...
1, 8, 2568, 5592968, 44418808968, 936239675880968, ...
1, 16, 26928, 515092048, 50363651248560, 16811849850663255376, ...
MAPLE
A:= (n, k)-> `if`(k=0, 1, ceil(2^(n-1))*add(add((-1)^i*
binomial(j, i)*binomial(j-i, n)^k, i=0..j), j=0..k*n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
A[n_, k_] := Sum[If[k == 0, 1, Binomial[j+n-1, n]^k] Sum[(-1)^(i-j)* Binomial[i, j], {i, j, n k}], {j, 0, n k}];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 04 2021 *)
PROG
(PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(j+n-1, n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020
CROSSREFS
Columns k=0..3 give: A000012, A011782, A052141, A316673.
Rows n=0..2 give: A000012, A000670, A059516.
Main diagonal gives A316677.
Sequence in context: A316564 A214742 A204124 * A101479 A136170 A245188
KEYWORD
nonn,tabl,walk
AUTHOR
Alois P. Heinz, Jul 10 2018
STATUS
approved

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Last modified June 15 19:43 EDT 2024. Contains 373410 sequences. (Running on oeis4.)