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

1,2

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

Cf. A001924, A026729.

Sequence in context: A128139 A208721 A208777 * A132108 A210489 A344821

Adjacent sequences: A104729 A104730 A104731 * A104733 A104734 A104735

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