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%I #18 May 09 2021 08:04:25
%S 1,1,2,2,7,14,6,34,86,168,24,209,648,1473,2840,120,1546,5752,14988,
%T 32344,61870,720,13327,58576,173007,414160,866695,1649232,5040,130922,
%U 671568,2228544,5876336,13373190,27422352,51988748
%N T(n, k) = n! * [x^n] exp(k * x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.
%F T(n, k) = (-1)^n*U(-n, 1, -k), where U is the Kummer U function.
%F T(n, k) = n! * L(n, -k), where L is the Laguerre polynomial function.
%F T(n, k) = n! * Sum_{j=0..n} binomial(n, j) * k^j / j!.
%e Triangle starts:
%e [0] 1;
%e [1] 1, 2;
%e [2] 2, 7, 14;
%e [3] 6, 34, 86, 168;
%e [4] 24, 209, 648, 1473, 2840;
%e [5] 120, 1546, 5752, 14988, 32344, 61870;
%e [6] 720, 13327, 58576, 173007, 414160, 866695, 1649232;
%e [7] 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748;
%e .
%e Array whose upward read antidiagonals are the rows of the triangle.
%e n\k 0 1 2 3 4 5
%e --------------------------------------------------------------------
%e [0] 1, 2, 14, 168, 2840, 61870, ...
%e [1] 1, 7, 86, 1473, 32344, 866695, ...
%e [2] 2, 34, 648, 14988, 414160, 13373190, ...
%e [3] 6, 209, 5752, 173007, 5876336, 224995745, ...
%e [4] 24, 1546, 58576, 2228544, 91356544, 4094022230, ...
%e [5] 120, 13327, 671568, 31636449, 1542401920, 80031878175, ...
%e [6] 720, 130922, 8546432, 490102164, 28075364096, 1671426609550, ...
%p # Rows of the array:
%p A := (n, k) -> (n + k)!*LaguerreL(n + k, -k):
%p seq(print(seq(simplify(A(n, k)), k = 0..6)), n = 0..6);
%p # Columns of the array:
%p egf := n -> exp(n*x/(1-x))/(1-x): ser := n -> series(egf(n), x, 16):
%p C := (k, n) -> (n + k)!*coeff(ser(k), x, n + k):
%p seq(print(seq(C(k, n), n = 0..6)), k=0..6);
%t T[n_, k_] := (-1)^(n) HypergeometricU[-n, 1, -k];
%t Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
%t (* Alternative: *)
%t T[n_, k_] := n ! LaguerreL[n , -k];
%t Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
%o (SageMath) # Columns of the array:
%o def column(k, len):
%o R.<x> = PowerSeriesRing(QQ, default_prec=len+k)
%o f = exp(k * x / (1 - x)) / (1 - x)
%o return f.egf_to_ogf().list()[k:]
%o for col in (0..6): print(column(col, 8))
%o # Alternative:
%o @cached_function
%o def L(n, x):
%o if n == 0: return 1
%o if n == 1: return 1 - x
%o return (L(n-1, x) * (2*n - 1 - x) - L(n-2, x)*(n - 1)) / n
%o A344048 = lambda n, k: factorial(n)*L(n, -k)
%o print(flatten([[A344048(n, k) for k in (0..n)] for n in (0..7)]))
%o (PARI)
%o T(n, k) = n! * sum(j=0, n, binomial(n, j) * k^j / j!)
%o for(n=0, 9, for(k=0, n, print(T(n, k))))
%Y T(n, n) = A277373(n). T(2*n, n) = A344049(n). Row sums are A343849.
%Y Cf. A343847.
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, May 08 2021