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A306547
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).
1
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
OFFSET
1,3
COMMENTS
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).
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215.
FORMULA
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.
EXAMPLE
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.
MATHEMATICA
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
PROG
(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)]
CROSSREFS
Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3), this sequence.
Sequence in context: A252158 A285996 A191618 * A143888 A016546 A303611
KEYWORD
sign,tabl
AUTHOR
G. C. Greubel, Feb 22 2019
STATUS
approved