OFFSET
0,2
LINKS
Paul D. Hanna, Table of rows 0..14 listed as n, a(n) for n = 0..7059
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).
EXAMPLE
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;...
PROG
(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(""))
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Dec 14 2008
STATUS
approved