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A276158 Triangle read by rows: T(n,k) = 6*k*(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.

%I A276158 #64 Jul 07 2022 02:30:22
%S A276158 1,2,6,3,12,12,4,18,24,18,5,24,36,36,24,6,30,48,54,48,30,7,36,60,72,
%T A276158 72,60,36,8,42,72,90,96,90,72,42,9,48,84,108,120,120,108,84,48,10,54,
%U A276158 96,126,144,150,144,126,96,54
%N A276158 Triangle read by rows: T(n,k) = 6*k*(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.
%C A276158 The row sums of the triangle provide the positive terms of A000578.
%C A276158 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).
%C A276158 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
%H A276158 Norman Whitehead, <a href="/A276158/a276158.png">Kaprekar Network Graph for 3 Digits</a>
%H A276158 Norman Whitehead, <a href="/A276158/a276158.py.txt">Kaprekar Network Graph Node Count Verification</a>
%F A276158 Sum_{k=0..n} T(n,k) = T(n,0)^3 = A000578(n+1).
%F A276158 G.f. as triangle: (1+4*x*y + x^2*y^2)/((1-x)^2*(1-x*y)^2). - _Robert Israel_, Aug 31 2016
%F A276158 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
%e A276158 Triangle starts:
%e A276158 ----------------------------------------------
%e A276158 n \ k |  0   1    2    3    4    5    6    7
%e A276158 ----------------------------------------------
%e A276158 0     |  1;
%e A276158 1     |  2,  6;
%e A276158 2     |  3, 12,  12;
%e A276158 3     |  4, 18,  24,  18;
%e A276158 4     |  5, 24,  36,  36,  24;
%e A276158 5     |  6, 30,  48,  54,  48,  30;
%e A276158 6     |  7, 36,  60,  72,  72,  60,  36;
%e A276158 7     |  8, 42,  72,  90,  96,  90,  72,  42;
%e A276158 ...
%p A276158 T:= (n, k) -> `if`(k=0, n+1, 6*k*(n+1-k)):
%p A276158 seq(seq(T(n, k), k=0..n), n=0..30); # _Robert Israel_, Aug 31 2016
%t A276158 Table[If[k == 0, n + 1, 6 k (n + 1 - k)], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 25 2016 *)
%o A276158 (PARI) T(n, k) = if (k==0, n+1, 6*k*(n+1-k));
%o A276158 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Aug 25 2016
%o A276158 (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
%o A276158 (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
%Y A276158 Cf. A000578, A008458, A276189.
%K A276158 nonn,tabl
%O A276158 0,2
%A A276158 _Stefano Maruelli_, Aug 22 2016
%E A276158 Corrected and rewritten by _Bruno Berselli_, Sep 01 2016