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A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

%I A144650 #25 Oct 14 2023 06:57:31
%S A144650 5,8,13,11,18,25,14,23,32,41,17,28,39,50,61,20,33,46,59,72,85,23,38,
%T A144650 53,68,83,98,113,26,43,60,77,94,111,128,145,29,48,67,86,105,124,143,
%U A144650 162,181,32,53,74,95,116,137,158,179,200,221,35,58,81,104,127,150,173,196,219,242,265
%N A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.
%C A144650 First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - _Vincenzo Librandi_, Nov 21 2012
%H A144650 Vincenzo Librandi, <a href="/A144650/b144650.txt">Rows n = 1..100, flattened</a>
%F A144650 Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - _R. J. Mathar_, Jan 15 2009, Jan 05 2011
%F A144650 From _G. C. Greubel_, Oct 14 2023: (Start)
%F A144650 T(n, n) = A001844(n).
%F A144650 T(n, n-1) = A001105(n), n >= 2.
%F A144650 T(n, n-2) = A142463(n-1), n >= 3.
%F A144650 T(n, n-3) = (-1)*A147973(n+2), n >= 4.
%F A144650 Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
%F A144650 G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)
%e A144650 Triangle begins:
%e A144650    5;
%e A144650    8, 13;
%e A144650   11, 18, 25;
%e A144650   14, 23, 32, 41;
%e A144650   17, 28, 39, 50,  61;
%e A144650   20, 33, 46, 59,  72,  85;
%e A144650   23, 38, 53, 68,  83,  98, 113;
%e A144650   26, 43, 60, 77,  94, 111, 128, 145;
%e A144650   29, 48, 67, 86, 105, 124, 143, 162, 181;
%e A144650   32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.
%t A144650 T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* _Vincenzo Librandi_, Nov 21 2012 *)
%o A144650 (Magma) [2*n*k + n + k + 1: k in [1..n], n in [1..11]]; // _Vincenzo Librandi_, Nov 21 2012
%o A144650 (SageMath) flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Oct 14 2023
%Y A144650 Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.
%Y A144650 Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).
%K A144650 nonn,tabl,easy
%O A144650 1,1
%A A144650 _Vincenzo Librandi_, Jan 13 2009