OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5050
FORMULA
Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
EXAMPLE
n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
1;
4, 3;
9, 8, 5;
16, 15, 12, 7;
25, 24, 21, 16, 9;
36, 35, 32, 27, 20, 11;
49, 48, 45, 40, 33, 24, 13;
64, 63, 60, 55, 48, 39, 28, 15;
81, 80, 77, 72, 65, 56, 45, 32, 17;
... etc. - Philippe Deléham, Mar 07 2013
MATHEMATICA
Table[n^2 - k^2, {n, 12}, {k, 0, n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)
PROG
(Magma) [n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024
(SageMath) flatten([[n^2-k^2 for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, Mar 12 2024
CROSSREFS
KEYWORD
AUTHOR
Reinhard Zumkeller, May 24 2004
STATUS
approved