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.
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
Examples
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, ...
Crossrefs
Programs
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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);
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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 -
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))))
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SageMath
# Columns of the array. def column(k, len): R.= 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))
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.