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A104732
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.
2
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
OFFSET
1,2
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).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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
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;
...
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, May 04 2008
MATHEMATICA
nn = 10; Map[Select[#, # > 0 &] &, Drop[CoefficientList[
Series[y x/(1 - x - y x + y x^3)/(1 - x), {x, 0, nn}], {x, y}],
1]] // Grid (* Geoffrey Critzer, Mar 17 2015 *)
PROG
(Python)
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]+list(map(iadd, a, b[:-1]))+[1]
c+=b
return c
# Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
CROSSREFS
Sequence in context: A128139 A208721 A208777 * A132108 A377000 A210489
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Mar 20 2005
EXTENSIONS
Edited by M. F. Hasler, Apr 26 2008
More terms from R. J. Mathar and Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
STATUS
approved