OFFSET
0,9
FORMULA
T(n, k) = n! * [x^n] Laguerre(k, log(1 - x)).
From Natalia L. Skirrow, Aug 27 2025: (Start)
D-finite with T(n,k) = (n-1)*T(n-1,k)+T(n,k-1)-(n-2)*T(n-1,k-1).
O.g.f.: hypergeom([1,y/(1-y)],[],x)/(1-y).
Row o.g.f.: (y/(1-y))_n/(1-y), where (x)_n is the Pochhammer symbol/rising factorial.
Row o.g.f. is also 0^n + y/(1-y)^(n+1)*Prod_{j=1..n-2}(j+1-j*y).
E.g.f.: 1/((1-y)*(1-x)^(y/(1-y))).
Column e.g.f.: hypergeom([-k],[1],log(1-y)).
T(n,k) = [x^k] (1+x)^k*(x)_n.
(End)
EXAMPLE
Array begins:
[0] 1, 1, 1, 1, 1, 1, 1, ...
[1] 0, 1, 2, 3, 4, 5, 6, ...
[2] 0, 1, 3, 6, 10, 15, 21, ...
[3] 0, 2, 7, 16, 30, 50, 77, ...
[4] 0, 6, 23, 57, 115, 205, 336, ...
[5] 0, 24, 98, 257, 546, 1021, 1750, ...
[6] 0, 120, 514, 1407, 3109, 6030, 10696, ...
[7] 0, 720, 3204, 9076, 20695, 41330, 75356, ...
[8] 0, 5040, 23148, 67456, 157865, 323005, 602517, ...
[9] 0, 40320, 190224, 567836, 1358564, 2837549, 5396650, ...
MAPLE
A := (n, k) -> add(binomial(k, j)*abs(Stirling1(n, j)), j = 0..n):
seq(seq(A(n-k, k), k = 0..n), n = 0..10);
# Expanding rows or columns:
RowSer := n -> series((1+x)^k*GAMMA(x + n)/GAMMA(x), x, 12):
Trow := n -> k -> coeff(RowSer(n), x, k):
ColSer := n -> series(orthopoly:-L(n, log(1 - x)), x, 12):
Tcol := k -> n -> n! * coeff(ColSer(k), x, n):
seq(lprint(seq(Trow(n)(k), k = 0..7)), n = 0..9);
seq(lprint(seq(Tcol(k)(n), n = 0..7)), k = 0..9);
PROG
(Python)
from functools import cache
@cache
def T(n: int, k: int) -> int:
if n == 0: return 1
if k == 0: return 0
return (n - 1) * T(n - 1, k) + T(n, k - 1) - (n - 2) * T(n - 1, k - 1)
for n in range(7): print([T(n, k) for k in range(7)])
CROSSREFS
Contains A271700 in transpose.
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Aug 27 2025
STATUS
approved