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A306547
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Triangle read by rows, defined by Riordan's general Eulerian recursion: T(n, k) = (k+3)*T(n-1, k) + (n-k-2) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).
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1
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1, 1, -2, 1, -11, 4, 1, -55, 35, -8, 1, -274, 210, -91, 16, 1, -1368, 986, -637, 219, -32, 1, -6837, 3180, -3473, 1752, -507, 64, 1, -34181, -1431, -17951, 10543, -4563, 1147, -128, 1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256, 1, -854494, -1726360, -1490890, -2314, -177832, 84176, -28105, 5627, -512
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OFFSET
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1,3
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COMMENTS
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Row sums are {1, -1, -6, -27, -138, -831, -5820, -46563, -419070, -4190703, ...}.
The Mathematica code for e(n,k,m) gives eleven sequences of which the first few are in the OEIS (see Crossrefs section).
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.
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LINKS
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FORMULA
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T(n, k) = (k+3)*T(n-1, k) + (n-k-2)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-2)^(n-1).
e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
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EXAMPLE
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Triangle begins with:
1.
1, -2.
1, -11, 4.
1, -55, 35, -8.
1, -274, 210, -91, 16.
1, -1368, 986, -637, 219, -32.
1, -6837, 3180, -3473, 1752, -507, 64.
1, -34181, -1431, -17951, 10543, -4563, 1147, -128.
1, -170900, -145310, -129950, 48442, -30524, 11470, -2555, 256.
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MATHEMATICA
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e[n_, 0, m_]:= 1; (* Example for m=3 *)
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k, 0, n-1}], {n, 1, 10}]], {m, 0, 10}]
T[n_, 1]:= 1; T[n_, n_]:= (-2)^(n-1); T[n_, k_]:= T[n, k] = (k+3)*T[n-1, k] + (n-k-2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten
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PROG
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(PARI) {T(n, k) = if(k==1, 1, if(k==n, (-2)^(n-1), (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)))};
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")))
(Sage)
def T(n, k):
if (k==1): return 1
elif (k==n): return (-2)^(n-1)
else: return (k+3)*T(n-1, k) + (n-k-2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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