A141431 - OEIS

%I #25 Mar 31 2021 04:10:30

%S 0,0,5,0,8,14,0,11,20,27,0,14,26,36,44,0,17,32,45,56,65,0,20,38,54,68,

%T 80,90,0,23,44,63,80,95,108,119,0,26,50,72,92,110,126,140,152,0,29,56,

%U 81,104,125,144,161,176,189,0,32,62,90,116,140,162,182,200,216,230,0,35,68,99,128,155,180,203,224,243,260,275

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

%H G. C. Greubel, <a href="/A141431/b141431.txt">Rows n = 1..50 of the triangle, flattened</a>

%F G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = y^2*x*(x*y-4*y+x+2)/((1-y)^3*(1-x)^2). - _R. J. Mathar_, Nov 27 2015. x and y swapped to align with standard, 19 Feb 2020

%F Sum_{k=1..n} T(n, k) = (n-1)*n*(7*n+1)/6 = A245301(n-1). - _G. C. Greubel_, Mar 31 2021

%e Triangle begins as:

%e   0;

%e   0,  5;

%e   0,  8, 14;

%e   0, 11, 20, 27;

%e   0, 14, 26, 36,  44;

%e   0, 17, 32, 45,  56,  65;

%e   0, 20, 38, 54,  68,  80,  90;

%e   0, 23, 44, 63,  80,  95, 108, 119;

%e   0, 26, 50, 72,  92, 110, 126, 140, 152;

%e   0, 29, 56, 81, 104, 125, 144, 161, 176, 189;

%p A141431 := proc(n,k)

%p         (k-1)*(3*n-k+1) ;

%p end proc:

%p seq(seq(A141431(n,k),k=1..n),n=1..14) ; # _R. J. Mathar_, Nov 10 2011

%t Table[(k-1)*(3*n-k+1), {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 31 2021 *)

%o (Magma) [(k-1)*(3*n-k+1): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 31 2021

%o (Sage) flatten([[(k-1)*(3*n-k+1) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 31 2021

%Y Columns: A016789 (k=2), A016933 (k=3), A008591 (k=4).

%Y Cf. A245301 (row sums).

%K nonn,easy,tabl

%O 1,3

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

%E More terms added by _G. C. Greubel_, Mar 31 2021