Wikipedia:Reference desk/Archives/Mathematics/2015 April 1

= April 1 =

Adding numbers
If you have columns of numbers: # |  k1 |  k2 |  k3 |  k4 |  k5 |  k6  | # 0| 1  |  1  |  1  |  1  |  1  |  1  | # 1|  1  |  2  |  2  |  2  |  2  |  2  | # 2|  1  |  3  |  4  |  4  |  4  |  4  | # 3|  1  |  4  |  7  |  8  |  8  |  8  | # 4|  1  |  5  |  11 |  15 |  16 |  16 | # 5|  1  |  6  |  16 |  26 |  31 |  32 | # 6|  1  |  7  |  22 |  42 |  57 |  63 | # 7|  1  |  8  |  29 |  64 |  99 |  120| # 8|  1  |  9  |  37 |  93 |  163|  219| # 9|  1  |  10 |  46 |  130|  256|  382|   n|  n^0|  n  |(n^2+n+2)/2| (n^3+5n+6)/6|  ? | ?  | Where a number equals to the number below it plus the one on its bottom right. I heard somewhere you add the equations or something. Can you come up with a formula where you input n to get the number for each column? Is there a generalization for all columns? K as the column and n as the row. But most importantly, how do you create equations like this? — Preceding unsigned comment added by Someone with a Question (talk • contribs) 12:48, 1 April 2015 (UTC)
 * First of all, your description seems to be upside-down: each cell in your diagram is the sum of cell above it and the cell on its top left; e.g. the 16 in the fifth row is 5 + 11. This sort of thing is called a recurrence relation, though that article has very little to say on multi-variable RRs, which is what your problem is. I found this question on StackExchange which discusses a method of generating a  formula for any cell in the grid (the example there is essentially Pascal's triangle, where the rule is that the value of a cell is the sum of the two cells above and to the left of it. (Googling "recurrence relation two variables" found this, and lots of similar hits.) AndrewWTaylor (talk) 14:30, 1 April 2015 (UTC)


 * The value in column k of row n is the coefficient of x^(k-1) in the formal power series expansion of
 * $$\frac{(1+x)^n}{1-x}$$
 * (assuming column numbering starts at 1). Alternatively, it is the sum of the first k values in the nth row of Pascal's triangle if k <= n, or 2^n otherwise. Gandalf61 (talk) 14:47, 1 April 2015 (UTC)


 * I believe this is . --RDBury (talk) 16:11, 1 April 2015 (UTC)