# Counts of Binomial Coefficients

Paul D. Hanna pauldhanna at juno.com
Wed Sep 29 04:28:49 CEST 2004

```Consider the sequences:
Numbers that appear as binomial coefficients exactly k times.

Could someone extend the sequences A06,A08,A10 below (if possible)?
There does not appear to be any terms for odd k > 3.
Are there terms for even k > 8?

Also I would  like to know if there is a formula for A04 below ...
it is infinite since it includes all triangular numbers.

I label 7 sequences below and discuss them briefly.
My program ran for n = 1..5000000, and here are the results.

(A03) Numbers that appear as binomial coefficients exactly 3 times.
(A04) Numbers that appear as binomial coefficients exactly 4 times.
(A05) Numbers that appear as binomial coefficients exactly 5 times.
(A06) Numbers that appear as binomial coefficients exactly 6 times.
(A08) Numbers that appear as binomial coefficients exactly 8 times.
(A10) Numbers that appear as binomial coefficients exactly 10 times.

Sequence A03 = A000984 (Central binomial coefficients)
except for first 2 terms: {1,2}.

Sequence A04 begins:
10,15,21,28,35,36,45,55,56,66,78,84,91,105,126,136,153,165,
171,190,220,231,253,276,286,300,325,330,351,364,378,406,435,
455,462,465,495,496,528,560,561,595,630,666,680,703,715,741,
780,792,816,820,861,903,946,969,990,1001, ... infinite # of terms ...

Sequence A05 - appears to have no terms.

Sequence A06 - please extend - begins:
120,210,1540,7140,11628,24310,

Sequence A08 - please extend - begins:
3003,

Sequence A10 - what are the terms (if any)?

Perhaps there are no more terms for A06, A08, A10?