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A371026
Triangle read by rows: T(n, k) = 4^n*Sum_{j=0..k} (-1)^(k - j)*binomial(k, j)* Pochhammer(j/4, n).
3
1, 0, 1, 0, 5, 2, 0, 45, 30, 6, 0, 585, 510, 180, 24, 0, 9945, 10350, 4950, 1200, 120, 0, 208845, 247590, 144900, 48600, 9000, 720, 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040, 0, 151412625, 216093150, 164872260, 80713080, 26334000, 5594400, 705600, 40320
OFFSET
0,5
FORMULA
T(n, k) = k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = n! for n >= 0. - Werner Schulte, Mar 17 2024
EXAMPLE
Triangle read by rows:
[0] 1;
[1] 0, 1;
[2] 0, 5, 2;
[3] 0, 45, 30, 6;
[4] 0, 585, 510, 180, 24;
[5] 0, 9945, 10350, 4950, 1200, 120;
[6] 0, 208845, 247590, 144900, 48600, 9000, 720;
[7] 0, 5221125, 6855030, 4655070, 1940400, 504000, 75600, 5040;
MAPLE
A371026 := (n, k) -> local j; 4^n*add((-1)^(k - j)*binomial(k, j)*pochhammer(j/4, n), j = 0..k): seq(seq(A371026(n, k), k = 0..n), n = 0..9);
PROG
(Python)
from functools import cache
@cache
def T(n, k): # After Werner Schulte
if k == 0: return 0**n
if k == n: return n * T(n-1, n-1)
return k * T(n-1, k-1) + (4*n - 4 + k) * T(n-1, k)
for n in range(8): print([T(n, k) for k in range(n + 1)])
# Peter Luschny, Mar 17 2024
CROSSREFS
Cf. A000142 (main diagonal), A007696 (column 1), A371027 (row sums).
Cf. A371025.
Sequence in context: A319231 A058512 A111560 * A324609 A211991 A354596
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 08 2024
STATUS
approved