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A343847
T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.
2
1, 1, 1, 2, 2, 1, 6, 7, 3, 1, 24, 34, 14, 4, 1, 120, 209, 86, 23, 5, 1, 720, 1546, 648, 168, 34, 6, 1, 5040, 13327, 5752, 1473, 286, 47, 7, 1, 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1, 362880, 1441729, 671568, 173007, 32344, 4929, 654, 79, 9, 1
OFFSET
0,4
FORMULA
T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.
EXAMPLE
Triangle starts:
0: 1;
1: 1, 1;
2: 2, 2, 1;
3: 6, 7, 3, 1;
4: 24, 34, 14, 4, 1;
5: 120, 209, 86, 23, 5, 1;
6: 720, 1546, 648, 168, 34, 6, 1;
7: 5040, 13327, 5752, 1473, 286, 47, 7, 1;
8: 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k 0 1 2 3 4 5 6
-----------------------------------------------------------------
0: 1, 1, 1, 1, 1, 1, 1, ...
1: 1, 2, 3, 4, 5, 6, 7, ...
2: 2, 7, 14, 23, 34, 47, 62, ...
3: 6, 34, 86, 168, 286, 446, 654, ...
4: 24, 209, 648, 1473, 2840, 4929, 7944, ...
5: 120, 1546, 5752, 14988, 32344, 61870, 108696, ...
6: 720, 13327, 58576, 173007, 414160, 866695, 1649232, ...
7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...
MAPLE
T := proc(n, k) option remember;
if n = k then return 1 elif n = k+1 then return k+1 fi;
(2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
seq(print(seq(T(n , k), k = 0..n)), n = 0..7);
MATHEMATICA
T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Alternative: *)
TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
PROG
(PARI)
T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
for(n=0, 9, for(k=0, n, print(T(n, k))))
(SageMath) # Columns of the array.
def column(k, len):
R.<x> = PowerSeriesRing(QQ, default_prec=len)
f = exp(k * x / (1 - x)) / (1 - x)
return f.egf_to_ogf().list()
for col in (0..6): print(column(col, 20))
CROSSREFS
Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
Cf. A021009 (Laguerre polynomials), A344048.
Sequence in context: A141712 A098539 A222073 * A135880 A077873 A123305
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 07 2021
STATUS
approved