login
A085484
Symmetric square array, read by antidiagonals: T(k, k) = T(0, k + 1) = Sum_{m = 0..k} C(k, m)*T(m, k - m) for k >= 0; T(0, 0) = 1; T(n, k) = T(n - 1, k) + T(n, k - 1) for n, k >= 1.
3
1, 1, 1, 2, 2, 2, 8, 4, 4, 8, 40, 12, 8, 12, 40, 224, 52, 20, 20, 52, 224, 1368, 276, 72, 40, 72, 276, 1368, 9008, 1644, 348, 112, 112, 348, 1644, 9008, 63488, 10652, 1992, 460, 224, 460, 1992, 10652, 63488, 476160, 74140, 12644, 2452, 684, 684, 2452, 12644, 74140, 476160
OFFSET
0,4
COMMENTS
The main diagonal is equal to the first row shifted left.
Antidiagonal sums give A085486. First row is A085485; table is symmetric under transpose, so that first column equals the first row. Second row gives partial sums of first row.
EXAMPLE
Rows begin:
1 1 2 8 40 224 1368 9008 ...
1 2 4 12 52 276 1644 10652 ...
2 4 8 20 72 348 1992 12644 ...
8 12 20 40 112 460 2452 15096 ...
40 52 72 112 224 684 3136 18232 ...
224 276 348 460 684 1368 4504 22736 ...
1368 1644 1992 2452 3136 4504 9008 31744 ...
9008 10652 12644 15096 18232 22736 31744 63488 ...
63488 74140 86784 101880 120112 142848 174592 238080 ...
MAPLE
A := proc(n, k) option remember;
if n = 0 then
1
elif n > 2*k then
A(n, n-k)
elif k = n then
add(binomial(n-1, i) * A(n-1, i), i = 0 .. n - 1)
else
A(n-1, k)+A(n-1, k-1)
end if
end proc:
for n from 0 to 6 do seq(A(n+k, k), k=0..12) od; # Yu-Sheng Chang, Jan 16 2020
PROG
(PARI) A85484=Map(); A085484(n, k)={if(n>k, [n, k]=[k, n], !k, return(1), n==k, n=!k++); mapisdefined(A85484, [n, k])|| mapput(A85484, [n, k], if(n, A085484(n-1, k)+A085484(n, k-1), sum(m=0, k-1, binomial(k-1, m)*A085484(m, k-1-m)))); mapget(A85484, [n, k])} \\ M. F. Hasler, Feb 17 2020
CROSSREFS
Cf. A085485 (first row and diagonal), A085486 (antidiagonal sums).
Sequence in context: A240284 A344897 A011202 * A326480 A116585 A230935
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 02 2003
EXTENSIONS
a(17)-a(18) corrected by Yu-Sheng Chang, Jan 16 2020
Name edited by M. F. Hasler, Feb 17 2020
STATUS
approved