A141433 - OEIS

%I #23 Apr 01 2021 17:27:19

%S 0,0,4,0,7,12,0,10,18,24,0,13,24,33,40,0,16,30,42,52,60,0,19,36,51,64,

%T 75,84,0,22,42,60,76,90,102,112,0,25,48,69,88,105,120,133,144,0,28,54,

%U 78,100,120,138,154,168,180

%N Triangle T(n, k) = (k-1)*(3*n-k), read by rows.

%H G. C. Greubel, <a href="/A141433/b141433.txt">Rows n = 1..50 of the triangle, flattened</a>

%F G.f.: Sum_{k>=0} Sum_{n>=0} T(n,k)*x^n*y^k = y^2*x*(2*x+1-3*y)/((1-y)^3*(x-1)^2).  (G.f. for the full array, not just the triangular subspace) - _R. J. Mathar_, Feb 19 2020

%F Sum_{k=1..n} T(n, k) = A304993(n-1) = (n-1)*n*(7*n -2)/6. - _G. C. Greubel_, Apr 01 2021

%e Triangle begins as:

%e   0;

%e   0,  4;

%e   0,  7, 12;

%e   0, 10, 18, 24;

%e   0, 13, 24, 33,  40;

%e   0, 16, 30, 42,  52,  60;

%e   0, 19, 36, 51,  64,  75,  84;

%e   0, 22, 42, 60,  76,  90, 102, 112;

%e   0, 25, 48, 69,  88, 105, 120, 133, 144;

%e   0, 28, 54, 78, 100, 120, 138, 154, 168, 180;

%p A141433 := proc(n,m) (m-1)*(3*n-m) ; end proc:

%p seq(seq(A141433(n,m),m=1..n),n=1..18) ; # _R. J. Mathar_, Sep 14 2011

%t Flatten[Table[(m-1)(3n-m),{n,10},{m,n}]] (* _Harvey P. Dale_, Feb 04 2016 *)

%o (Magma) [(k-1)*(3*n-k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 01 2021

%o (Sage) flatten([[(k-1)*(3*n-k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 01 2021

%Y Cf. A304993 (row sums).

%K nonn,easy,tabl

%O 1,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 06 2008