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A222403
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Triangle read by rows: left and right edges are A000217, interior entries are filled in using the Pascal triangle rule.
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5
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0, 1, 1, 3, 2, 3, 6, 5, 5, 6, 10, 11, 10, 11, 10, 15, 21, 21, 21, 21, 15, 21, 36, 42, 42, 42, 36, 21, 28, 57, 78, 84, 84, 78, 57, 28, 36, 85, 135, 162, 168, 162, 135, 85, 36, 45, 121, 220, 297, 330, 330, 297, 220, 121, 45, 55, 166, 341, 517, 627, 660, 627, 517, 341, 166, 55
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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In general, if the sequence defining the left and right edges is [a_0, a_1, ...], the row sums [s_0, s_1, ...] are given by s_0=a_0 and, for n>0,
s_n = 2a_n + Sum_{i=1..n-1} 2^(n-i) a_i.
Conversely, given the rows sums [s_0, s_1, ...], the edge sequence is [a_0, a_1, ...] where a_0=s_0 and, for n>0, a_n = (s_n - Sum_{i=1..n-1} s_i)/2.
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LINKS
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FORMULA
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G.f. as triangle: (1+x-4*x*y+x*y^2+x^2*y^2)*y/((1-y)^2*(-x*y+1)^2*(-x*y-y+1)). - Robert Israel, Apr 04 2018
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EXAMPLE
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Triangle begins:
0
1, 1
3, 2, 3
6, 5, 5, 6
10, 11, 10, 11, 10
15, 21, 21, 21, 21, 15
21, 36, 42, 42, 42, 36, 21
28, 57, 78, 84, 84, 78, 57, 28
...
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MAPLE
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d:=[seq(n*(n+1)/2, n=0..14)];
f:=proc(d) local T, M, n, i;
M:=nops(d);
T:=Array(0..M-1, 0..M-1);
for n from 0 to M-1 do T[n, 0]:=d[n+1]; T[n, n]:=d[n+1]; od:
for n from 2 to M-1 do
for i from 1 to n-1 do T[n, i]:=T[n-1, i-1]+T[n-1, i]; od: od:
lprint("triangle:");
for n from 0 to M-1 do lprint(seq(T[n, i], i=0..n)); od:
lprint("row sums:");
lprint([seq( add(T[i, j], j=0..i), i=0..M-1)]);
end;
f(d);
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MATHEMATICA
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t[n_, n_] := n*(n+1)/2; t[n_, 0] := n*(n+1)/2; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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