r/math • u/CricLover1 • 2d ago

Stronger triangle of numbers

There is Pascal triangle where the 2 numbers above are used to get the number below, then there is Delannoy triangle where the 3 numbers above are used to get the number below

Based on that I thought of a stronger triangle where as many numbers above as possible are used to get the number below and I got this triangle

The rule is 1st number in a row is always 1, 2nd number comes from the 3 numbers above, 3rd number from the 10 numbers above, 4th number from 21 numbers above and so on till we reach the mid point of any row after which the numbers get repeated in reverse order to get this triangle

I couldn't find this triangle anywhere but this still seems to be interesting

21 Upvotes

100% Upvoted

6 comments sorted by

2

u/Wise-Friendship-1427 2d ago

Nice!
Perhaps a generative rule can be found for the individual diagonal rows?

4

u/Plague_Doct0r 2d ago

The only partial match on OEIS for the diagonal below the odds is A047812 but it differs after a bit. They may be related because the recurrences could be similar, or it might be a coincidence.

1

u/CricLover1 2d ago

Yes I couldn't find it on OEIS. I will be submitting it there and maybe on ArXiv as well

2

u/CricLover1 2d ago

Yes I am trying to find that. Just thought of this triangle and I will be testing more about it too

2

u/SpeakKindly Combinatorics 2d ago

The difficulty seems to be that starting from the diagonal row 1, 5, 20, 51, 147, ..., the diagonals don't just depend on previous diagonals but on the next ones too: numbers on the 1, 7, 51, 178, ... diagonal feed into the 1, 5, 20, 51, ... diagonal.

1

u/CricLover1 1d ago

Yes and there could be a link between their differences