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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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FORMULA
| 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
| 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
| 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 (mathar(AT)strw.leidenuniv.nl), May 04 2008
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PROG
| Python program from Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008. Replace leading dots by spaces.
.def A104732_rows(n):
........"""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]+map(iadd, a, b[:-1])+[1]
................c+=b
........return c
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CROSSREFS
| Cf. A001924, A026729.
Sequence in context: A188002 A186974 A128139 * A132108 A125175 A193376
Adjacent sequences: A104729 A104730 A104731 * A104733 A104734 A104735
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 20 2005
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EXTENSIONS
| Edited by M. F. Hasler (www.univ-ag.fr/~mhasler), Apr 26 2008
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
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