A003238 
Number of rooted trees with n vertices in which vertices at the same level have the same degree.

very sparce, only clicky noise up to the end. sounds like someone spilled a box of legos in 8bit sound

Residue counts:
2   [97, 103]
3   [67, 73, 60]
4   [51, 50, 46, 53]
5   [49, 52, 24, 33, 42]
6   [40, 44, 28, 27, 29, 32]
7   [30, 26, 25, 29, 26, 32, 32]
8   [29, 29, 22, 27, 22, 21, 24, 26]
9   [16, 28, 14, 21, 16, 24, 30, 29, 22]
10   [24, 30, 13, 17, 22, 25, 22, 11, 16, 20]
11   [23, 9, 18, 22, 20, 21, 17, 18, 20, 13, 19]
12   [21, 21, 13, 14, 15, 16, 19, 23, 15, 13, 14, 16]

Density information:
A(x) at x=10,100,1000,...:
[7, 18, 40, 76, 130, 200]

A(x)/x at x=10,100,1000,...:
0.7,0.18,0.04,0.0076,0.0013,0.0002,

First 100 first differences:
[0, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 13, 1, 12, 8, 16, 1, 26, 1, 29, 13, 28, 1, 51, 6, 42, 19, 56, 1, 87, 1, 77, 29, 79, 16, 134, 1, 106, 43, 145, 1, 195, 1, 178, 77, 177, 1, 288, 11, 253, 80, 278, 1, 379, 32, 361, 107, 352, 1, 573, 1, 440, 163, 516, 46, 699, 1, 627, 178, 701, 1, 961, 1, 776, 288, 884, 37, 1164, 1, 1130, 294, 1071, 1, 1589, 83, 1267, 353, 1455, 1, 1976, 51, 1701, 441, 1700, 110, 2426, 1, 2040, 561, 2359]

First 99 second differences:
1,0,1,-1,3,-3,4,-2,4,-6,12,-12,11,-4,8,-15,25,-25,28,-16,15,-27,50,-45,36,-23,37,-55,86,-86,76,-48,50,-63,118,-133,105,-63,102,-144,194,-194,177,-101,100,-176,287,-277,242,-173,198,-277,378,-347,329,-254,245,-351,572,-572,439,-277,353,-470,653,-698,626,-449,523,-700,960,-960,775,-488,596,-847,1127,-1163,1129,-836,777,-1070,1588,-1506,1184,-914,1102,-1454,1975,-1925,1650,-1260,1259,-1590,2316,-2425,2039,-1479,1798,

The formula for this seq is a(n)= 1 + sum(a(i)) where i's are integers such that n=1 modulo i, and i != 1. (According to OEIS)
I have no idea why the formula wouldn't be written as a(n)= sum(a(i)) where i's are integers such that n=1 modulo i
since a(1) = 1.
This is a strictly increasing sequence since for all n, n=1 mod 1 and n=1 mod (n-1) thus a(n) >= a(1)+a(n-1) which implies a(n)>a(n-1)

Proof of this formula goes something like this: 
condition: is two nodes have the same level(depth), they have the same degree
A(n) is the set of trees with n nodes that satisfy the condition aforementioned. by def, a(n)=|A(n)|.
for any A(n), we can categorize the types of such trees by the degree of the root of the tree, k.
if k = 1 we can think of adding a new root to the graphs of A(n-1).
if k = n-1, we can think of a star-shaped rooted tree, which all other nodes are directly connected to the root. 
when k is not a divisor of n-1, we cannot satisfy the condition since in that case in the level before the max level, there must be at least one tree and one non-tree.
if k is a divisor of n-1, say n-1=sk then any graph of A(s) can be copied k times can connect to the root giving def(root)=k and since these are all the same trees that satisfy the condition, this new tree must satisfy the condition too. 

Jayme Kim