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
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, ...
Programs
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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);
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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))))
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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!.