A152547 Triangle, read by rows, derived from Pascal's triangle (see g.f. and example for generating methods).

1, 2, 3, 1, 4, 2, 2, 5, 3, 3, 3, 1, 1, 6, 4, 4, 4, 4, 2, 2, 2, 2, 2, 7, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 8, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 9, 7, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Author

Paul D. Hanna, Dec 14 2008

Keywords

• nonn
• tabf

Examples

			The number of terms in row n is C(n,[n/2]).
Triangle begins:
[1],
[2],
[3,1],
[4,2,2],
[5,3,3,3,1,1],
[6,4,4,4,4,2,2,2,2,2],
[7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1],
[8,6,6,6,6,6,6,4,4,4,4,4,4,4,4,4,4,4,4,4,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2],
[9,7,7,7,7,7,7,7,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1],
...
ILLUSTRATION OF GENERATING METHOD.
Row n is derived from the binomial coefficients in the following way.
Place markers in an array so that the number of contiguous markers
in row k is C(n,k) and then count the markers along columns.
For example, row 6 of this triangle is generated from C(6,k) like so:
------------------------------------------
1: o - - - - - - - - - - - - - - - - - - -
6: o o o o o o - - - - - - - - - - - - - -
15:o o o o o o o o o o o o o o o - - - - -
20:o o o o o o o o o o o o o o o o o o o o
15:o o o o o o o o o o o o o o o - - - - -
6: o o o o o o - - - - - - - - - - - - - -
1: o - - - - - - - - - - - - - - - - - - -
------------------------------------------
Counting the markers along the columns gives row 6 of this triangle:
[7,5,5,5,5,5,3,3,3,3,3,3,3,3,3,1,1,1,1,1].
Continuing in this way generates all the rows of this triangle.
...
Number of repeated terms in each row of this triangle forms A008315:
1;
1;
1, 1;
1, 2;
1, 3, 2;
1, 4, 5;
1, 5, 9, 5;
1, 6, 14, 14;
1, 7, 20, 28, 14;...
		

Links

• Paul D. Hanna, Table of rows 0..14 listed as n, a(n) for n = 0..7059

Crossrefs

Cf. A152548 (row squared sums), A008315; A152545.

Programs

PARI

{T(n,k)=polcoeff(sum(j=0,n,(x^binomial(n,j) - 1)/(x-1)),k)}
for(n=0,10, for(k=0, binomial(n,n\2)-1, print1(T(n,k),","));print(""))

Formula

G.f. of row n: Sum_{k=0..n} (x^binomial(n,k) - 1)/(x-1) = Sum_{k=0..binomial(n,n\2)-1} T(n,k)*x^k.

A152548(n) = Sum_{k=0..C(n,[n/2])-1} T(n,k)^2 = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).