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A104732
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Square array T[i,j]=T[i-1,j]+T[i-1,j-1], T[1,j]=j, T[i,1]=1, read by antidiagonals.
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2
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1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 8, 5, 1, 6, 9, 12, 12, 6, 1, 7, 11, 16, 20, 17, 7, 1, 8, 13, 20, 28, 32, 23, 8, 1, 9, 15, 24, 36, 48, 49, 30, 9, 1, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 11, 19, 32, 52, 80, 112, 129, 102, 47, 11, 1, 12, 21, 36, 60, 96, 144, 192, 201, 140, 57, 12, 1
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OFFSET
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1,2
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COMMENTS
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Original definition was "Triangle, row sums are A001924". Reading the rows of the triangle as antidiagonals of a square array allows a precise, yet simple definition and a method for computing the terms. - M. F. Hasler, Apr 26 2008
When formatted as a triangle, row sums are A001924: 1, 3, 7, 14, 26...(apply the partial sum operator twice to the Fibonacci sequence).
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LINKS
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FORMULA
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The triangle is extracted from A * B; where A = [1; 2, 1; 3, 2, 1;...], B = [1; 0, 1; 0, 1, 1; 0, 0, 2, 1;...]; both infinite lower triangular matrices with the rest of the terms zeros. The sequence in "B" (1, 0, 1, 0, 1, 1, 0, 0, 2, 1...) = A026729.
As a square array, g.f. Sum T[i,j] x^j y^i = xy/((1-(1+x)y)*(1-x)^2). - Alec Mihailovs (alec(AT)mihailovs.com), Apr 26 2008
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EXAMPLE
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The first few rows of the triangle (= rising diagonals of the square array) are:
1;
2, 1;
3, 3, 1;
4, 5, 4, 1;
5, 7, 8, 5, 1;
6, 9, 12, 12, 6, 1;
...
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MAPLE
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A104732 := proc(i, j) coeftayl(coeftayl(x*y/(1-x)^2/(1-y*(1+x)), y=0, i), x=0, j) ; end: for d from 1 to 20 do for j from d to 1 by -1 do printf("%d, ", A104732(d-j+1, j)) ; od: od: # R. J. Mathar, May 04 2008
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MATHEMATICA
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nn = 10; Map[Select[#, # > 0 &] &, Drop[CoefficientList[
Series[y x/(1 - x - y x + y x^3)/(1 - x), {x, 0, nn}], {x, y}],
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PROG
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(Python)
"""Produces n rows of A104732 triangle"""
from operator import iadd
a, b, c = [], [1], [1]
for i in range(2, n+1):
a, b = b, [i]+list(map(iadd, a, b[:-1]))+[1]
c+=b
return c
# Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from R. J. Mathar and Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
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STATUS
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approved
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