cp's OEIS Frontend

A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

%I A143037 #16 Aug 17 2025 21:45:37
%S A143037 1,3,4,6,9,10,10,16,19,20,15,25,31,34,35,21,36,46,52,55,56,28,49,64,
%T A143037 74,80,83,84,36,64,85,100,110,116,119,120,45,81,109,130,145,155,161,
%U A143037 164,165,55,100,136,164,185,200,210,216,219,220
%N A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).
%C A143037 Right border = tetrahedral numbers, left border = triangular numbers.
%C A143037 Alternatively this is the square array A(n,k)
%C A143037    1,   3,   6,  10,  15,  21,  28,  36,  45,  55, ...
%C A143037    4,   9,  16,  25,  36,  49,  64,  81, 100, 121, ...
%C A143037   10,  19,  31,  46,  64,  85, 109, 136, 166, 199, ...
%C A143037   20,  34,  52,  74, 100, 130, 164, 202, 244, 290, ...
%C A143037   35,  55,  80, 110, 145, 185, 230, 280, 335, 395, ...
%C A143037   56,  83, 116, 155, 200, 251, 308, 371, 440, 515, ...
%C A143037   ...
%C A143037 read by antidiagonals where A(n,k) = n*(n^2 + 3*k*n + 3*k^2 - 1)/6 is the sum of n triangular numbers starting at A000217(k). - _R. J. Mathar_, May 06 2015
%F A143037 T(n,k) = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6. - _R. J. Mathar_, Aug 31 2022
%e A143037 First few rows of the triangle:
%e A143037    1;
%e A143037    3,  4;
%e A143037    6,  9, 10;
%e A143037   10, 16, 19,  20;
%e A143037   15, 25, 31,  34,  35;
%e A143037   21, 36, 46,  52,  55,  56;
%e A143037   28, 49, 64,  74,  80,  83,  84;
%e A143037   36, 64, 85, 100, 110, 116, 119, 120;
%e A143037   ...
%p A143037 A143037 := proc(n,k)
%p A143037     k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6 ;
%p A143037 end proc:
%p A143037 seq(seq(A143037(n,k),k=1..n),n=1..12) ; # _R. J. Mathar_, Aug 31 2022
%t A143037 T[n_,k_] = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6;Table[T[n,k],{n,10},{k,n}]//Flatten (* _James C. McMahon_, Aug 13 2025 *)
%Y A143037 Cf. A001296 (row sums).
%K A143037 nonn,tabl,easy
%O A143037 1,2
%A A143037 _Gary W. Adamson_ & _Roger L. Bagula_, Jul 18 2008