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A362277
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j * binomial(n-j,j)/(n-j)!.
9
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -2, 1, 1, 1, -3, -8, 1, 6, 1, 1, 1, -4, -11, 10, 41, 16, 1, 1, 1, -5, -14, 25, 106, 31, -20, 1, 1, 1, -6, -17, 46, 201, -44, -461, -132, 1, 1, 1, -7, -20, 73, 326, -299, -1952, -895, 28, 1, 1, 1, -8, -23, 106, 481, -824, -5123, -1028, 6481, 1216, 1
OFFSET
0,14
LINKS
FORMULA
E.g.f. of column k: exp(x - k*x^2/2).
T(n,k) = T(n-1,k) - k*(n-1)*T(n-2,k) for n > 1.
T(n,k) = n! * Sum_{j=0..floor(n/2)} (-k/2)^j / (j! * (n-2*j)!).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, ...
1, -2, -5, -8, -11, -14, -17, ...
1, -2, 1, 10, 25, 46, 73, ...
1, 6, 41, 106, 201, 326, 481, ...
1, 16, 31, -44, -299, -824, -1709, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n\2, (-k/2)^j/(j!*(n-2*j)!));
CROSSREFS
Columns k=0..6 give A000012, (-1)^n * A001464(n), A293604, A362278, A362176, A362279, A362177.
Main diagonal gives A362276.
T(n,2*n) gives A362282.
Sequence in context: A287641 A265312 A241531 * A367955 A273894 A308035
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Apr 13 2023
STATUS
approved