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.

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.

Links

Table of n, a(n) for n=0..54.

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

Adjacent sequences: A085481 A085482 A085483 * A085485 A085486 A085487

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