A344048 T(n, k) = n! * [x^n] exp(k * x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 2, 2, 7, 14, 6, 34, 86, 168, 24, 209, 648, 1473, 2840, 120, 1546, 5752, 14988, 32344, 61870, 720, 13327, 58576, 173007, 414160, 866695, 1649232, 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748

Offset: 0

Author

Peter Luschny, May 08 2021

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]    1;
[1]    1,      2;
[2]    2,      7,     14;
[3]    6,     34,     86,     168;
[4]   24,    209,    648,    1473,    2840;
[5]  120,   1546,   5752,   14988,   32344,    61870;
[6]  720,  13327,  58576,  173007,  414160,   866695,  1649232;
[7] 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748;
.
Array whose upward read antidiagonals are the rows of the triangle.
n\k   0       1        2          3           4              5
--------------------------------------------------------------------
[0]   1,      2,      14,       168,        2840,         61870, ...
[1]   1,      7,      86,      1473,       32344,        866695, ...
[2]   2,     34,     648,     14988,      414160,      13373190, ...
[3]   6,    209,    5752,    173007,     5876336,     224995745, ...
[4]  24,   1546,   58576,   2228544,    91356544,    4094022230, ...
[5] 120,  13327,  671568,  31636449,  1542401920,   80031878175, ...
[6] 720, 130922, 8546432, 490102164, 28075364096, 1671426609550, ...
		

Crossrefs

T(n, n) = A277373(n). T(2*n, n) = A344049(n). Row sums are A343849.

Cf. A343847.

Programs

Maple

# Rows of the array:
A := (n, k) -> (n + k)!*LaguerreL(n + k, -k):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n = 0..6);
# Columns of the array:
egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 16):
C := (k, n) -> (n + k)!*coeff(ser(k), x, n + k):
seq(print(seq(C(k, n), n = 0..6)), k=0..6);

Mathematica

T[n_, k_] := (-1)^(n) HypergeometricU[-n, 1,  -k];
Table[T[n, k], {n, 0, 7}, {k, 0, n}]  // Flatten
(* Alternative: *)
T[n_, k_] := n ! LaguerreL[n , -k];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten

PARI

T(n, k) = n! * sum(j=0, n, binomial(n, 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. = PowerSeriesRing(QQ, default_prec=len+k)
    f = exp(k * x / (1 - x)) / (1 - x)
    return f.egf_to_ogf().list()[k:]
for col in (0..6): print(column(col, 8))
# Alternative:
@cached_function
def L(n, x):
    if n == 0: return 1
    if n == 1: return 1 - x
    return (L(n-1, x) * (2*n - 1 - x) - L(n-2, x)*(n - 1)) / n
A344048 = lambda n, k: factorial(n)*L(n, -k)
print(flatten([[A344048(n, k) for k in (0..n)] for n in (0..7)]))

Formula

T(n, k) = (-1)^n*U(-n, 1, -k), where U is the Kummer U function.

T(n, k) = n! * L(n, -k), where L is the Laguerre polynomial function.

T(n, k) = n! * Sum_{j=0..n} binomial(n, j) * k^j / j!.