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Triangle read by rows: T(x, y) = 0 if y > x, = 1 if y = 0, or = 2*Sum_{k >= 1, x-k^2 >= y} T(x-k^2, y-1) otherwise. The zeros are omitted from the sequence.
2

%I #8 Jan 21 2013 09:58:11

%S 1,1,2,1,2,4,1,2,4,8,1,4,4,8,16,1,4,12,8,16,32,1,4,12,32,16,32,64,1,4,

%T 12,32,80,32,64,128,1,4,16,32,80,192,64,128,256,1,6,16,56,80,192,448,

%U 128,256,512,1,6,24,56,176,192,448,1024,256,512,1024

%N Triangle read by rows: T(x, y) = 0 if y > x, = 1 if y = 0, or = 2*Sum_{k >= 1, x-k^2 >= y} T(x-k^2, y-1) otherwise. The zeros are omitted from the sequence.

%C Comments from R. J. Mathar, Oct 31 2006:

%C This sequence provides the seeds for the construction of columns (vertical recurrence) of A122510 insofar as each row of A123937 provides two sides of auxiliary arrays b(.,.,.) from which a column of A122510 emerges as the third side:

%C A122510(d,n)=b(0,d,n) [with an auxiliary, virtual A122510(0,n)=1].

%C Seeds to construct two sides of b(.,.,.):

%C b(x,0,n)=A123937(n,x) for x<=n; b(n,y,n)=A123937(n,n) for y>=0.

%C Recurrence within the b(.,.,.) : b(x,y,n)=b(x,y-1,n)+b(x+1,y-1,n) for x<n.

%C Graphical support as if the array were built top-down and left-to-right from the seeds:

%C Triangle stump ("stump" means cut-off/finiteness at the bottom and top)

%C ...................b(n,0,n)...b(n,1,n)...b(n,2,n)....

%C ..............................

%C .............b(2,0,n)...b(2,1,n)....

%C .........b(1,0,n)...b(1,1,n)....

%C ...b(0,0,n)..b(0,1,n)...b(0,2,n)....

%C equals triangle stump (note that the top line is constant) T(x,y)=A123937(x,y)

%C ...................T(n,n)...T(n,n)...T(n,n)....

%C ..............................

%C .............T(n,2).....b(2,1,n)....

%C .........T(n,1).....b(1,1,n)....

%C ...T(n,0)....b(0,1,n)...b(0,2,n)....

%C equals triangle stump

%C ...................T(n,n)...T(n,n)...T(n,n)....

%C ..............................

%C .............T(n,2).....b(2,1,n)....

%C .........T(n,1).....b(1,1,n)....

%C ...T(n,0)...A122510(1,n).A122510(2,n).A122510(3,n)....

%e Triangle begins:

%e 1

%e 1 2

%e 1 2 4

%e 1 2 4 8

%e 1 4 4 8 16

%e 1 4 12 8 16 32

%e 1 4 12 32 16 32 64

%e 1 4 12 32 80 32 64 128

%K nonn,tabl

%O 0,3

%A _David W. Wilson_, Oct 30 2006