Hypertribonacci number array has 6 existing OEIS seqs as rows and columns

[Jonathan Post]

If we construct hypertribonacci numbers, which are to the
hyperfibonacci array of A136431 as the tribonacci numbers A000073 are
to the Fibonacci numbers A000045, then we have 6 existing OEIS
sequences as rows and columns (or 9 if we include the all 0 seq, the
all 1 seq, and n).

The recursion is:

a(k,n) = apply partial sum operator k times to tribonacci numbers A000073.

========================================
n=0..|.1.|.2.|...3.|...4.|.....5.|.......6.|.......7.|.........8.|..........9.|......10.|
========================================
k=0..|.0.|.0.|...1.|...2.|.....4.|.......7.|.....13.|.......24.|.......44.|.......81.|
A000073
k=1..|.0.|.0.|...2.|...4.|.....8.|.....15.|.....28.|.......52.|.......96.|.....177.|
A008937
k=2..|.0.|.0.|...3.|...7.|...15.|.....30.|.....58.|.....110.|.....206.|.....383.|
A062544
k=3..|.0.|.0.|...4.|.11.|...26.|.....56.|...114.|.....224.|.....430.|.....813.|
new
k=4..|.0.|.0.|...5.|.16.|...42.|.....98.|...212.|.....436.|.....866.|...1679.|
new
k=5..|.0.|.0.|...6.|.22.|...64.|...162.|...374.|.....810.|...1676.|...3355.| new
k=6..|.0.|.0.|...7.|.29.|...93.|...255.|...659.|...1469.|...3145.|...6500.| new
k=7..|.0.|.0.|...8.|.37.|.130.|...385.|.1044.|...2513.|...5658.|.12158.| new
k=8..|.0.|.0.|...9.|.46.|.176.|...561.|.1605.|...4118.|...9776.|.21934.| new
k=9..|.0.|.0.|.10.|.56.|.232.|...793.|.2398.|...6516.|.16292.|.38226.| new
k=10|.0.|.0.|.11.|.67.|.299.|.1092.|.3490.|.10006.|.16292.|.26298.| new
k=11|.0.|.0.|.12.|.79.|.378.|.1470.|.4960.|.14966.|.31258.|.57556.| new
========================================

It appears that:
n=4 column == A000124 Central polygonal numbers n(n+1)/2+1;
n=5 column == A000125 Cake numbers C(n+1,3)+n+1;
n=6 column == A055795 Binomial(m,4)+Binomial(n,2).

Is any of this valid, or is my full time substitute teaching of
teenagers plus night classes at teacher's college causing lack of
sleep which makes me see patterns that are spurious?

Thnaks for the attention and consideration.

-- Jonathan Vos Post