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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 (list; table; graph; refs; listen; history; text; internal format)
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.
LINKS
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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)