cp's OEIS Frontend

A174066 Irregular triangle, row sums = A000041, left border = A174065.

%I A174066 #6 Mar 12 2015 20:08:19
%S A174066 1,1,1,1,2,1,3,1,1,4,2,1,5,3,1,2,7,4,2,2,9,5,3,2,3,12,7,4,4,3,15,9,5,
%T A174066 6,3,4,19,12,7,8,6,4,25,15,9,10,9,4,5,31,19,12,14,12,8,5,38,25,15,18,
%U A174066 15,12,5,7,48,31,19,24,21,16,10,7,60,38,25,30,27,20,15,7,9,73,48,31,38,36
%N A174066 Irregular triangle, row sums = A000041, left border = A174065.
%C A174066 Left border = A174065: (1, 1, 1, 2, 3, 4, 5, 7, 9, 12,...) * its aerated variant (1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7,...) = A000041, the partition sequence: (1, 1, 2, 3, 5, 7, 11, 15, 22, 30,...).
%F A174066 The triangle is the result of three rules after beginning (1, 1, 1, 1,...):
%F A174066 Columns >1 are shifted down twice from previous columns; column terms = left border * (left border placed as a heading row); and row sums = A000041, the partition numbers. The rules force the next missing term in the triangle to be the leftmost term in column 1. This is found by taking p(n) for row n, then subtracting the sum of row n terms (minus leftmost term).
%e A174066 Heading and first few rows of the triangle =
%e A174066 .1,...1,...1,...2,...3,...4,...5,...7,...9,... = A174065.
%e A174066 .1;........................................... = .. 1 (A000041)
%e A174066 .1;........................................... = .. 1
%e A174066 .1,...1;...................................... = .. 2
%e A174066 .2,...1;...................................... = .. 3
%e A174066 .3,...1,...1;................................. = .. 5
%e A174066 .4,...2,...1;................................. = .. 7
%e A174066 .5,...3,...1,...2;............................ = ..11
%e A174066 .7,...4,...2,...2;............................ = ..15
%e A174066 .9....5,...3,...2,...3;....................... = ..22
%e A174066 .12,..7,...4,...4,...3;....................... = ..30
%e A174066 .15,..9,...5,...6,...3;...4;.................. = ..42
%e A174066 .19,.12,...7,...8,...6,...4;.................. = ..56
%e A174066 .25,.15,...9,..10,...9,...4,...5;............. = ..77
%e A174066 .31,.19,..12,..14,..12,...8,...5;............. = .101
%e A174066 .38,.25,..15,..18,..15,..12,...5,...7;........ = .135
%e A174066 .48,.31,..19,..24,..21,..16,..10,...7;........ = .176
%e A174066 .60,.38,..25,..30,..27,..20,..15,...7,...9;... = .231
%e A174066 .73,.48,..31,..38,..36,..28,..20,..14,...9;... = .297
%e A174066 ...
%e A174066 Example: leftmost term in 8th row has to be 7 = (15 - (4 + 2 + 2)); so we
%e A174066 place a 7 as next term in the heading, then multiply * leftmost column.
%e A174066 Finally, shift the columns down twice.
%Y A174066 Cf. A000041, A174065, A174067.
%K A174066 nonn,tabf
%O A174066 1,5
%A A174066 _Gary W. Adamson_, Mar 06 2010