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A106828
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Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).
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9
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1, 0, 0, 1, 0, 2, 0, 6, 3, 0, 24, 20, 0, 120, 130, 15, 0, 720, 924, 210, 0, 5040, 7308, 2380, 105, 0, 40320, 64224, 26432, 2520, 0, 362880, 623376, 303660, 44100, 945, 0, 3628800, 6636960, 3678840, 705320, 34650, 0, 39916800, 76998240, 47324376, 11098780, 866250, 10395
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OFFSET
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0,6
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COMMENTS
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Another version of the triangle is in A008306.
A signed version of this triangle is given by the exponential Riordan array [1, log(1+t)-t]. Its row sums are (-1)^n*(1-n). Another version is [1, log(1-t)+t], whose row sums are 1-n. - Paul Barry, May 10 2008
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j), where E2 are the second-order Eulerian numbers (A008517). - Peter Luschny, Jan 13 2016
Also the Bell transform of the sequence g(k) = k! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016
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EXAMPLE
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Rows 0 though 7 are:
1;
0,
0, 1;
0, 2,
0, 6, 3;
0, 24, 20,
0, 120, 130, 15;
0, 720, 924, 210;
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MAPLE
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A106828 := (n, k) -> add(binomial(j, n-2*k)*combinat:-eulerian2(n-k, j), j=0..n-k):
seq(print(seq(A106828(n, k), k=0..iquo(n, 2))), n=0..9); # Peter Luschny, Apr 20 2011 (revised Jan 13 2016)
# Second program, after David Callan in A008306:
A106828 := proc(n, k) option remember; if k = 0 then k^n elif k = 1 then (n-1)! elif n <= 2*k-1 then 0 else (n-1)*(procname(n-1, k) + procname(n-2, k-1)) fi end: seq((seq(A106828(n, k), k = 0..iquo(n, 2))), n=0..12); # Peter Luschny, Aug 24 2021
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MATHEMATICA
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Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n - 1, k] (k + 1) + Eulerian2[n - 1, k - 1] (2 n - k - 1)]];
T[n_, k_] := Sum[Binomial[j, n - 2 k] Eulerian2[n - k, j], {j, 0, n - k}];
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PROG
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(Haskell)
a106828 n k = a106828_tabf !! n !! k
a106828_row n = a106828_tabf !! n
a106828_tabf = map (fst . fst) $ iterate f (([1], [0]), 1) where
f ((us, vs), x) =
((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
(Sage) # uses[bell_transform from A264428]
# Computes the full triangle 0<=k<=n.
g = lambda k: factorial(k) if k>0 else 0
s = [g(k) for k in (0..n)]
return bell_transform(n, s)
(PARI) E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
T(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ Michel Marcus, Dec 07 2021
(Python)
from math import factorial
return k**n if k == 0 else factorial(n-1) if k == 1 else 0 if n <= 2*k - 1 else (n - 1)*(A106828(n-1, k) + A106828(n-2, k-1))
for n in range(14): print([A106828(n, k) for k in range(n//2 + 1)])
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CROSSREFS
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KEYWORD
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tabf,nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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