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Triangle in which row n is derived from (1,2,3,...,n) dot (n,n-1,...,1) with additive carryovers.
3

%I #12 Apr 14 2013 01:42:47

%S 1,2,4,3,7,10,4,10,16,20,5,13,22,30,35,6,16,28,40,50,56,7,19,34,50,65,

%T 77,84,8,22,40,60,80,98,112,120,9,25,46,70,95,119,140,156,165,10,28,

%U 52,80,110,140,168,192,210,220

%N Triangle in which row n is derived from (1,2,3,...,n) dot (n,n-1,...,1) with additive carryovers.

%C Carryovers (additive) are defined as "add current product to next product". For example: (1,2,3) dot (3,2,1) with carryovers = ((1*3=3 ), (2*2+3=7), (1*3+7=10), so row 2 = (3, 7, 10).

%C Row sums = A002415: (1, 6, 20, 50, 105, 196,...)

%F By rows, (1,2,3,...) dot (...3,2,1); add current product to next product.

%F As an array, row 0 = the tetrahedral numbers, (1, 4, 10, 20, 35,...). n-th row adds n*(1, 3, 6, 10, 15,...) termwise.

%e Row 2 = (3, 7, 10) = (1, 2, 3) dot (3, 2, 1) with carryovers, thus: (3 = 1*3; 7 = 2*2 + 3; 10 = 3*1 + 7.

%e First few rows of the array =

%e 1,...4,..10,..20,..35,...

%e 2,...7,..16,..30,..50,...

%e 3,..10,..22,..40,..65,...

%e ...

%e Example: row 1 is obtained by adding (1, 3, 6, 10, 15,...) termwise to (1, 4, 10, 20, 35,...).

%e First few rows of the triangle =

%e .

%e 1;

%e 2, 4;

%e 3, 7, 10;

%e 4, 10, 16, 20;

%e 5, 13, 22, 30, 35;

%e 6, 16, 28, 40, 50, 56;

%e 7, 19, 34, 50, 65, 77, 84;

%e 8, 22, 40, 60, 80, 98, 112, 120;

%e 9, 25, 46, 70, 95, 119, 140, 156, 165;

%e 10, 28, 52 80, 110, 140, 168, 192, 210, 220;

%e ...

%Y Cf. A002415

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Dec 13 2010