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A373050
Triangle read by rows: T(n, k) = (Sum_{i=0..n-k} (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) for 0 <= k <= n.
0
1, 2, 1, 7, 6, 2, 32, 39, 24, 6, 181, 284, 252, 120, 24, 1214, 2325, 2680, 1860, 720, 120, 9403, 21234, 30030, 27240, 15480, 5040, 720, 82508, 214459, 358848, 400890, 299040, 143640, 40320, 5040, 808393, 2375736, 4586456, 6077904, 5599440, 3541440, 1471680, 362880, 40320
OFFSET
0,2
FORMULA
T(n, k) = n * (T(n-1, k-1) + T(n-1, k)) for 0 < k < n with initial values T(n, 0) = A000153(n+1) and T(n, n) = A000142(n).
E.g.f. of column k: (exp(-t) / (1-t)^3) * (t / (1-t))^k.
E.g.f.: exp(x * t / (1-t) - t) / (1-t)^3.
EXAMPLE
Triangle T(n, k) starts:
n\k : 0 1 2 3 4 5 6 7 8
=========================================================================
0 : 1
1 : 2 1
2 : 7 6 2
3 : 32 39 24 6
4 : 181 284 252 120 24
5 : 1214 2325 2680 1860 720 120
6 : 9403 21234 30030 27240 15480 5040 720
7 : 82508 214459 358848 400890 299040 143640 40320 5040
8 : 808393 2375736 4586456 6077904 5599440 3541440 1471680 362880 40320
etc.
PROG
(PARI) T(n, k) = { sum(i=0, n-k, (-1)^i * binomial(n-k, i) * (n+2-i)!) * binomial(n, k) / ((k+1) * (k+2)) }
CROSSREFS
Cf. A000153 (column 0), A000142 (main diagonal and 1st subdiagonal).
Cf. A000255 (alt. row sums).
Sequence in context: A039814 A178120 A180568 * A248950 A329058 A078301
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, May 20 2024
STATUS
approved