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.

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

Author

Gary W. Adamson, Mar 20 2005

Keywords

• nonn
• tabl

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).

Examples

			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;
...
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001924, A026729.

Programs

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 *)

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

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

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