[seqfan] An overlooked family of sequences

[Joe Slater]

Many existing sequences are part of an interesting family that seems to
have gone unnoticed.

Define the flattened binomial of two numbers p and q such that the n'th
flattened binomial is p^(n-1), p^(n-2)*q, ... p*q^(n-2), q^(n-1). For
instance, the 4th flattened binomial of (2,3) is (8,12,18,27). The
flattened binomial sum is the sum of these values, in this case 65.

For convenience, let us denote the n'th value of the flattened binomial sum
for (p,q) as FBS^n(p,q)

We can consequently define our function as follows:
For n>0; p,q >= 0
FBS^n(p,q) = (p^n-q^n)/(p-q) when p<>q
FBS^n(p,q) = n*p^(n-1) when p=q

With this definition we can obviously produce an infinite number of
sequences. Some of the existing ones (without regard for the first term)
are:
a : a(n) = FBS^n(0,0) is A000007, powers of 0
a : a(n) = FBS^n(0,1) is A000012, powers of 1
a : a(n) = FBS^n(1,1) is A000027, the positive integers
a : a(n) = FBS^n(0,2) is A000079, powers of 2
a : a(n) = FBS^n(1,2) is A000225
a : a(n) = FBS^n(2,2) is A001787
a : a(n) = FBS^n(0,3) is A000244, powers of 3
a : a(n) = FBS^n(1,3) is A003462
a : a(n) = FBS^n(2,3) is A001047
a : a(n) = FBS^n(3,3) is A027471
a : a(n) = FBS^n(0,4) is A000302, powers of 4
a : a(n) = FBS^n(1,4) is A002450
a : a(n) = FBS^n(2,4) is A006516
a : a(n) = FBS^n(3,4) is A005061
a : a(n )= FBS^n(4,4) is A002697
a : a(n) = FBS^n(0,5) is A000351 powers of 5
a : a(n) = FBS^n(1,5) is A003463
a : a(n) = FBS^n(2,5) is A016127
a : a(n) = FBS^n(3,5) is A005059
a : a(n) = FBS^n(4,5) is A005060
a : a(n) = FBS^n(5,5) is A053464
a : a(n) = FBS^n(0,6) is A000400 powers of 6
a : a(n) = FBS^n(1,6) is A003464
a : a(n) = FBS^n(2,6) is A016129
a : a(n) = FBS^n(3,6) is A016137
a : a(n) = FBS^n(4,6) is A016149  (and also A081199, which is essentially
the same)
a : a(n) = FBS^n(5,6) is A005062
a : a(n) = FBS^n(6,6) is A053469

I stopped checking there, but it would be interesting to identify the first
FBS sequence *not* found in the OEIS. I'm not surprised that such a simple
mathematical relationship should appear in so many sequences, but it *is*
surprising that it hasn't  been explicitly identified as such.

Incidentally, the first, second, etc. members of each of these sequences
form their own super-sequences: the first members are (1,1,1 ...), which is
the trivial sequence A000012; if we skip the first three lines the second
members are (2,3,4,4,5,6,5,6,7,8 ...) which is A108872, the "sums of
ordinal references for a triangular table read by columns, top to bottom";
the third members are A215631; the fourth members are A321490; later
super-sequences don't seem to have been added to the OEIS.

Joe Slater

// P.S.
// PARI/GP code to produce a vector containing a generic flattened binomial
and to supply a flattened binomial sum:
flatbinomial(n,x='x,y='y) = vector(n,i,x^(n-i)*y^(i-1))
fbs(n=1,p=1,q=1) = vecsum(flatbinomial(n,p,q))