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A177878 Triangle in which row n is generated from (1,2,3,...,n) dot (n, n-1,...,1) with subtractive carryovers.

%I A177878 #16 Feb 09 2025 02:45:34
%S A177878 1,2,0,3,1,2,4,2,4,0,5,3,6,2,3,6,4,8,4,6,0,7,5,10,6,9,3,4,8,6,12,8,12,
%T A177878 6,8,0,9,7,14,10,15,9,12,4,5,10,8,16,12,18,12,16,8,10,0,11,9,18,14,21,
%U A177878 15,20,12,15,5,6,12,10,20,16,24,18,24,16,20,10,12,0
%N A177878 Triangle in which row n is generated from (1,2,3,...,n) dot (n, n-1,...,1) with subtractive carryovers.
%C A177878 The subtractive carryover dot product of two vectors (a(1),a(2),...,a(n)) dot (b(1),b(2),...,b(n)) = (c(1),...,c(n)) is defined by c(1) = a(1)*b(1) and c(i) = a(i)*b(i)-c(i-1), i>1.
%C A177878 A177877 = analogous triangle with additive carryovers.
%C A177878 A160770 = the analogous triangle using the triangular series as the generating vector.
%F A177878 By rows, dot product of (1,2,3,...) and (...3,2,1) with subtractive carryovers; such that current row product subtracts previous product.
%e A177878 Row 3 = (4, 2, 4, 0) = (1, 2, 3, 4) dot (4, 3, 2, 1) with subtractive carryovers = (4), then (2*3 - 4 = 2), (3*2 - 2 = 4), and (4*1 - 4 = 0).
%e A177878 First few rows of the triangle:
%e A177878   1;
%e A177878   2, 0;
%e A177878   3, 1, 2;
%e A177878   4, 2, 4, 0;
%e A177878   5, 3, 6, 2, 3;
%e A177878   6, 4, 8, 4, 6, 0;
%e A177878   7, 5, 10, 6, 9, 3, 4;
%e A177878   8, 6, 12, 8, 12, 6, 8, 0;
%e A177878   9, 7, 14, 10, 15, 9, 12, 4, 5;
%e A177878   10, 8, 16, 12, 18, 12, 16, 8, 10, 0;
%e A177878   11, 9, 18, 14, 21, 15, 20, 12, 15, 5, 6;
%e A177878   12, 10, 20, 16, 24, 18, 24, 16, 20, 10, 12, 0;
%e A177878   ...
%Y A177878 Cf. A005993 (row sums), A177877, A160770
%K A177878 nonn,tabl
%O A177878 0,2
%A A177878 _Gary W. Adamson_, Dec 13 2010