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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)[reply]

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)[reply]

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)[reply]

I believe this is OEIS: A052509. --RDBury (talk) 16:11, 1 April 2015 (UTC)[reply]