|
Search: id:A026374
|
|
|
| A026374 |
|
Triangular array T read by rows: T(n,0)=T(n,n)=1 for all n >= 0, T(n,k)=T(n-1,k-1) + T(n-1,k) for odd n and 1<=k<=n-1, T(n,k)=T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1<=k<=n-1. |
|
+0
18
|
|
| 1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 7, 17, 17, 7, 1, 1, 9, 30, 45, 30, 9, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1
(list; table; graph; listen)
|
|
|
OFFSET
|
0,5
|
|
|
COMMENT
|
T(n,k) is number of lattice paths from (0,0) to (n,n-2k) using steps U=(1,1), D=(1,-1) and, at levels ...-4,-2,0,2,4,..., also H=(2,0). Example: T(4,1)=6 because we have the following paths from (0,0) to (4,2): UUUD, UUH, UUDU, UDUU, HUU and DUUU. Row sums yield A026383. Column 1 is A032766, column 2 is A026381, column 3 is A026382. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2004
|
|
FORMULA
|
T(n, k) = number of integer strings s(0), ..., s(n) such that s(0)=0, s(n)=n-2k, where, for 1<=i<=n, s(i) is even if i is even and |s(i)-s(i-1)|<=1.
T(2n, k)=sum(3^(2j-k)*binomial(n, j)binomial(j, k-j), j=ceil(k/2)..k); T(2n+1, k)=T(2n, k-1)+T(2n, k). G.f.=(1+z+tz)/[1-(1+3t+t^2)z^2]=1+(1+t)z+(1+3t+t^2)z^2+... . Generating polynomial for row 2n is (1+3t+t^2)^n and for row 2n+1 it is (1+t)(1+3t+t^2)^n. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2004
T(2n, k)=sum(3^(2j-k)*binomial(n, j)*binomial(j, k-j), j=ceil(k/2)..k); T(2n+1, k)=T(2n, k-1)+T(2n, k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 30 2004
|
|
CROSSREFS
|
Cf. A026383.
Sequence in context: A136482 A026648 A026747 this_sequence A102716 A134510 A124020
Adjacent sequences: A026371 A026372 A026373 this_sequence A026375 A026376 A026377
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Clark Kimberling (ck6(AT)evansville.edu)
|
|
|
Search completed in 0.002 seconds
|
