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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.

%I A104732 #23 Apr 26 2020 15:20:31
%S A104732 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,
%T A104732 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,
%U A104732 11,19,32,52,80,112,129,102,47,11,1,12,21,36,60,96,144,192,201,140,57,12,1
%N 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.
%C A104732 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
%C A104732 When formatted as a triangle, row sums are A001924: 1, 3, 7, 14, 26...(apply the partial sum operator twice to the Fibonacci sequence).
%H A104732 Charles R Greathouse IV, <a href="/A104732/b104732.txt">Table of n, a(n) for n = 1..10000</a>
%F A104732 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.
%F A104732 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
%e A104732 The first few rows of the triangle (= rising diagonals of the square array) are:
%e A104732 1;
%e A104732 2, 1;
%e A104732 3, 3, 1;
%e A104732 4, 5, 4, 1;
%e A104732 5, 7, 8, 5, 1;
%e A104732 6, 9, 12, 12, 6, 1;
%e A104732 ...
%p A104732 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
%t A104732 nn = 10; Map[Select[#, # > 0 &] &,Drop[CoefficientList[
%t A104732     Series[y x/(1 - x - y x + y x^3)/(1 - x), {x, 0, nn}], {x, y}],
%t A104732 1]] // Grid (* _Geoffrey Critzer_, Mar 17 2015 *)
%o A104732 (Python)
%o A104732 def A104732_rows(n):
%o A104732     """Produces n rows of A104732 triangle"""
%o A104732     from operator import iadd
%o A104732     a,b,c = [], [1], [1]
%o A104732     for i in range(2,n+1):
%o A104732             a,b = b, [i]+list(map(iadd,a,b[:-1]))+[1]
%o A104732             c+=b
%o A104732     return c
%o A104732 # Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008
%Y A104732 Cf. A001924, A026729.
%K A104732 nonn,tabl
%O A104732 1,2
%A A104732 _Gary W. Adamson_, Mar 20 2005
%E A104732 Edited by _M. F. Hasler_, Apr 26 2008
%E A104732 More terms from _R. J. Mathar_ and Alec Mihailovs (alec(AT)mihailovs.com), May 04 2008