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A104345 Triangle read by rows: T(n,k) is the number of alternating permutations on [n+1] with 1 in position k+1, 0<=k<=n. 5
1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 5, 8, 6, 8, 5, 16, 25, 20, 20, 25, 16, 61, 96, 75, 80, 75, 96, 61, 272, 427, 336, 350, 350, 336, 427, 272, 1385, 2176, 1708, 1792, 1750, 1792, 1708, 2176, 1385, 7936, 12465, 9792, 10248, 10080, 10080, 10248, 9792, 12465, 7936 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
The mixed o.g.f./e.g.f. is Sum_{k=0..n} T(n, k)*x^n/n!*y^k = (sec(x) + tan(x))*(sec(x*y) + tan(x*y)).
T(n,k) = binomial(n,k)*A000111(k)*A000111(n-k). - Alois P. Heinz, Apr 25 2023
EXAMPLE
Table begins
\ k..0....1....2....3....4....
n
0 |..1
1 |..1....1
2 |..1....2....1
3 |..2....3....3....2
4 |..5....8....6....8....5
5 |.16...25...20...20...25...16
6 |.61...96...75...80...75...96...61
7 |272..427..336..350..350..336..427..272
For example, a(3,1) counts 2143, 3142, 4132---the alternating permutations on [4] with 1 in position 2.
MAPLE
b:= proc(u, o) option remember; `if`(u+o=0, 1,
add(b(o-1+j, u-j), j=1..u))
end:
T:= (n, k)-> binomial(n, k)*b(k, 0)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Apr 25 2023
CROSSREFS
Cf. A104346. Row sums are A001250; column k=0 and main diagonal are the up-down numbers (A000111); column k=1 is A065619.
T(2n,n) gives A362581.
Sequence in context: A342095 A340828 A123265 * A244516 A363264 A002339
KEYWORD
nonn,tabl
AUTHOR
David Callan, Mar 02 2005
STATUS
approved

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Last modified March 3 16:27 EST 2024. Contains 370512 sequences. (Running on oeis4.)