OFFSET
1,6
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..2390 (rows 1..30, flattened)
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 16.
M. Isachenkov, I. Kirsch, and V. Schomerus, Chiral Primaries in Strange Metals, arXiv preprint arXiv:1403.6857 [hep-th], 2014. See Eq. (4.6).
FORMULA
Sum_{k=0..floor(n^2/4)} T(n, k) = A000079(n-1) (row sums).
Sum_{k=0..(n+2 - ceiling(sqrt(4*n)))} T(n-k, k) = A002865(n+1) (antidiagonal sums).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A077957(n-1). - G. C. Greubel, Feb 13 2024
EXAMPLE
The differences between 5 3 4 3 1 and 4 2 2 yield row four : 1 1 2 3 1.
Triangle begins:
1;
1, 1;
1, 1, 2;
1, 1, 2, 3, 1;
1, 1, 2, 3, 5, 2, 2;
1, 1, 2, 3, 5, 7, 5, 4, 3, 1;
1, 1, 2, 3, 5, 7, 11, 8, 9, 7, 6, 2, 2;
1, 1, 2, 3, 5, 7, 11, 15, 14, 15, 15, 13, 11, 7, 4, 3, 1;
1, 1, 2, 3, 5, 7, 11, 15, 22, 21, 25, 25, 27, 23, 22, 15, 13, 8, 6, 2, 2;
...
MAPLE
MATHEMATICA
b[n_, k_] := b[n, k] = SeriesCoefficient[Sum[QBinomial[n, m, q], {m, 0, n}], {q, 0, k}];
T[n_, k_] := b[n, k] - b[n - 1, k];
Table[Table[T[n, k], {k, 0, n^2/4}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n, k, x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n, k | Coefficient(R!( (&+[qBinom(n, k, x): k in [0..n]]) ), k) >;
[A161161(n, k): k in [0..Floor(n^2/4)], n in [1..12]]; // G. C. Greubel, Feb 13 2024
(SageMath)
def t(n, k): # t = A083906
if k<0 or k> (n^2//4): return 0
elif n<2 : return n+1
else: return 2*t(n-1, k) - t(n-2, k) + t(n-2, k-n+1)
def A161161(n, k): return t(n, k) - t(n-1, k)
flatten([[A161161(n, k) for k in range(int(n^2//4)+1)] for n in range(1, 13)]) # G. C. Greubel, Feb 13 2024
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Alford Arnold, Jun 04 2009
STATUS
approved