1,5

I find it more useful to draw the table as columns instead of rows, where each column has 2^(column-1) rows. This table has a huge number of fascinating properties. I'll list a few here, but would like to add more: Each column shows most numbers in pairs, where the row of the paired number can be found by starting with the row number (starting with 0), converting to binary of a length determined by the column (power of 2), doing a one's compliment, and reversing the order of bits to determine the row number for the paired entry. So in the {1,13,6,13,2,6,2,1} column, row pairs are 001/011 (two 13's), 010/101 (two 6's), 000 / 111 (two 1's) and 100 / 110 (two 2's). In alternating columns, that leaves some "pairs" which map to themselves. The largest of these form a sequence, aka A048144. Graphed, subsequent columns show a fractal pattern. The rows of the table also convert neatly to another fascinating sequence, via the inverse stirling transform. Another conversion

(details later) describes the rows as a series of equations, the coefficients of which again embody this very H() table. (Amazing.) There are a huge number of simple relationships, such as every 4th term in a column is the sum of twice the number below it plus the one above. FYI, I figured this all out by calculating the table explicitly, and then staring at it for years. There's lots more to disclose, but I could use some guidance on how to proceed.

Brian Parsonnet (bparsonnet(AT)comcast.net), Mar 22 2008, Table of n, a(n) for n = 1..255

H(r,c) = sum of H(T(r),L(r)+j) * M(c-T(r)-1,j) for j = 0..c-L(r)-1, where M(y,z) = binomial distribution (y,z) when y - 1 > z, and (y,z)-1 when (y-1)<=z, and T(r) = A053645, and L(r) = A000523.

For 5 people, there are 9 target outcomes for which the last person gets his own name. But these 9 outcomes share only 4 possible probabilities, namely, 12/576, 8/576, 9/576, and 6/576. For the 9 outcomes, they distribute over these 4 probabilities with the distribution 1, 5, 2, 1 for a total likelihood of 76/576. As to the calculation of the table itself, look at H(3,5) which has the unique value of 73. The final step in the formula described is the dot product of two vectors: {0, 1, 5, 13} * {1, 4, 6, 3} = 73. {1,4,6,3} is almost the binomial distribution for 4 items, but the case (4,3) is 3 instead of 4 (as described by the formula).

Adjacent sequences: A136298 A136299 A136300 this_sequence A136302 A136303 A136304

Sequence in context: A085119 A010128 A029764 this_sequence A132690 A089086 A038631

uned,nonn,tabl

Brian Parsonnet (bparsonnet(AT)comcast.net), Mar 22 2008