OFFSET
0,2
COMMENTS
The row sums of the triangle provide the positive terms of A000578.
Similar triangles can be generated by the formula P(n,k,m) = (Q(k+1,m)-Q(k,m))*(n+1-k), where Q(i,r) = i^r-(i-1)^r, 0 < k <= n, and P(n,0,m) = n+1. T(n,k) is the case m=3, that is T(n,k) = P(n,k,3).
T(9,k) for 0 <= k <= 9 provides the indegrees of the 10 non-leaf nodes of the network graph of the Kaprekar Process on 3 digits when the nodes are listed in numerical order. Namely, nodes 000, 099, 198, 297, 396, 495, 594, 693, 792, and 891 have indegrees 10, 54, 96, 126, 144, 150, 144, 126, 96, 54, respectively. Result derived empirically. See "Kaprekar Network Graph for 3 Digits". - Norman Whitehead, May 16 2022
LINKS
Norman Whitehead, Kaprekar Network Graph for 3 Digits
Norman Whitehead, Kaprekar Network Graph Node Count Verification
FORMULA
Sum_{k=0..n} T(n,k) = T(n,0)^3 = A000578(n+1).
G.f. as triangle: (1+4*x*y + x^2*y^2)/((1-x)^2*(1-x*y)^2). - Robert Israel, Aug 31 2016
T(n,n-h) = (h+1)*A008458(n-h) for 0 <= h <= n. Therefore, the main diagonal of the triangle is A008458. - Bruno Berselli, Aug 31 2016
EXAMPLE
Triangle starts:
----------------------------------------------
n \ k | 0 1 2 3 4 5 6 7
----------------------------------------------
0 | 1;
1 | 2, 6;
2 | 3, 12, 12;
3 | 4, 18, 24, 18;
4 | 5, 24, 36, 36, 24;
5 | 6, 30, 48, 54, 48, 30;
6 | 7, 36, 60, 72, 72, 60, 36;
7 | 8, 42, 72, 90, 96, 90, 72, 42;
...
MAPLE
T:= (n, k) -> `if`(k=0, n+1, 6*k*(n+1-k)):
seq(seq(T(n, k), k=0..n), n=0..30); # Robert Israel, Aug 31 2016
MATHEMATICA
Table[If[k == 0, n + 1, 6 k (n + 1 - k)], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 25 2016 *)
PROG
(PARI) T(n, k) = if (k==0, n+1, 6*k*(n+1-k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 25 2016
(Magma) [IsZero(k) select n+1 else 6*k*(n+1-k): k in [0..n], n in [0..10]]; // Bruno Berselli, Aug 31 2016
(Magma) /* As triangle (see the second comment): */ m:=3; Q:=func<i, r | i^r-(i-1)^r>; P:=func<n, k, m | IsZero(k) select n+1 else (Q(k+1, m)-Q(k, m))*(n+1-k)>; [[P(n, k, m): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 31 2016
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Maruelli, Aug 22 2016
EXTENSIONS
Corrected and rewritten by Bruno Berselli, Sep 01 2016
STATUS
approved