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A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

%I A100345 #24 Jul 05 2025 14:29:16
%S A100345 0,1,2,4,6,8,9,12,15,18,16,20,24,28,32,25,30,35,40,45,50,36,42,48,54,
%T A100345 60,66,72,49,56,63,70,77,84,91,98,64,72,80,88,96,104,112,120,128,81,
%U A100345 90,99,108,117,126,135,144,153,162,100,110,120,130,140,150,160,170,180,190
%N A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.
%C A100345 Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - _Franklin T. Adams-Watters_, Apr 04 2010
%F A100345 T(n,0) = A000290(n).
%F A100345 T(n,1) = A002378(n) for n > 0.
%F A100345 T(n,2) = A005563(n) for n > 1.
%F A100345 T(n,3) = A028552(n) for n > 2.
%F A100345 T(n,4) = A028347(n+2) for n > 3.
%F A100345 T(n,5) = A028557(n) for n > 4.
%F A100345 T(n,6) = A028560(n) for n > 5.
%F A100345 T(n,7) = A028563(n) for n > 6.
%F A100345 T(n,8) = A028566(n) for n > 7.
%F A100345 T(n,9) = A028569(n) for n > 8.
%F A100345 T(n,10) = A098603(n) for n > 9.
%F A100345 T(n,n-5) = A071355(n-4) for n > 4.
%F A100345 T(n,n-4) = A054000(n-1) for n > 3.
%F A100345 T(n,n-3) = A014107(n) for n > 2.
%F A100345 T(n,n-2) = A046092(n-1) for n > 1.
%F A100345 T(n,n-1) = A000384(n) for n > 0.
%F A100345 T(n,n) = A001105(n).
%F A100345 Row sums give A085789 for n > 0.
%F A100345 G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Jul 03 2025
%e A100345 Triangle begins:
%e A100345    0
%e A100345    1   2
%e A100345    4   6   8
%e A100345    9  12  15  18
%e A100345   16  20  24  28  32
%e A100345   25  30  35  40  45  50
%e A100345   36  42  48  54  60  66  72
%e A100345   49  56  63  70  77  84  91  98
%e A100345   64  72  80  88  96 104 112 120 128
%t A100345 Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Aug 16 2018 *)
%o A100345 (PARI) row(n) = vector(n+1, k, n*(n+k-1)); \\ _Amiram Eldar_, May 09 2025
%Y A100345 Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.
%K A100345 nonn,tabl,easy
%O A100345 0,3
%A A100345 _Reinhard Zumkeller_, Nov 18 2004