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A157013
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Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).
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4
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1, 1, -1, 1, -4, 1, 1, -15, 5, -1, 1, -58, 10, -6, 1, 1, -229, -66, -26, 7, -1, 1, -912, -1017, -288, 23, -8, 1, 1, -3643, -8733, -4779, -415, -41, 9, -1, 1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1, 1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1
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OFFSET
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1,5
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COMMENTS
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Row sums are: {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202,...}.
This recursion set doesn't seem to produce the Eulerian 2nd A008517.
The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019
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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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G. C. Greubel, Rows n=1..100 of triangle, flattned
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FORMULA
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e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
T(n, k) = (k+2)*T(n-1, k) + (n-k-1)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). - G. C. Greubel, Feb 22 2019
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EXAMPLE
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Triangle begins with:
1.
1, -1.
1, -4, 1.
1, -15, 5, -1.
1, -58, 10, -6, 1.
1, -229, -66, -26, 7, -1.
1, -912, -1017, -288, 23, -8, 1.
1, -3643, -8733, -4779, -415, -41, 9, -1.
1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1.
1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1.
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MATHEMATICA
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e[n_, 0, m_]:= 1;
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_]:=(-1)^(n-1); T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] + (n-k-1)*T[n-1, k-1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)
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PROG
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(PARI) {T(n, k) = if(k==1, 1, if(k==n, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019
(Sage)
def T(n, k):
if (k==1): return 1
elif (k==n): return (-1)^(n-1)
else: return (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019
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CROSSREFS
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Cf. A008517.
Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), This Sequence (m=3).
Sequence in context: A058711 A202906 A177984 * A141724 A208956 A271705
Adjacent sequences: A157010 A157011 A157012 * A157014 A157015 A157016
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KEYWORD
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sign,tabl
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AUTHOR
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Roger L. Bagula, Feb 21 2009
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STATUS
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approved
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