[seqfan] Re: In the vein of 103314

[Chai Wah Wu]

The sixth row 1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547
doesn't correspond to any sequence in OEIS.

The seventh row 1,7, 28,84, ... appears to be A000579 (minus the first few
terms).

The eighth row appears to be A014820

The ninth row 1,9,45,163,477,1197,2764, does not appear to correspond to
any sequence in OEIS

It appears that for p prime, the p-th row corresponds to
binomial(k+p-1,p-1) for k = 0,1,2,3,....
which is the maximum possible (i.e. the number of combinations with
repetitions of k choices from p categories).

Chai Wah

----- Original message -----
From: Allan Wechsler <acwacw at gmail.com>
Sent by: "SeqFan" <seqfan-bounces at list.seqfan.eu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc:
Subject: [seqfan] Re: In the vein of 103314
Date: Thu, Dec 21, 2017 1:19 PM

We can create a similar table by counting the number of distinct sums of k
(not necessarily distinct) nth roots of 1; this makes sense for any
positive integer n and nonnegative integer k. The

The first row (n=1) is 1,1,1,1,1,1 ... (A000012)

The second row is 1,2,3,4,5,6 ... (A000027)

The third row is 1,3,6,10,15,21 ... (A000217, beheaded)

The fourth row is 1,4,9,16,25,36 ... (A000290, beheaded)

The fifth row is 1,5,15,35,70,126 ... (I think this is a tail of A000332)

The sequence we have been calculating is the diagonal n=k, which is off the
main diagonal in the table above because n is 1-origin and k is 0-origin. I
believe that each row is polynomial with degree A000010(n) (Euler's totient
function phi(n)).