A108350 Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 32, 21, 6, 1, 1, 7, 31, 65, 65, 31, 7, 1, 1, 8, 43, 116, 161, 116, 43, 8, 1, 1, 9, 57, 189, 341, 341, 189, 57, 9, 1, 1, 10, 73, 288, 645, 842, 645, 288, 73, 10, 1, 1, 11, 91, 417, 1121, 1827, 1827, 1121, 417, 91

Offset: 0

Author

Paul Barry, May 31 2005

Keywords

• easy
• nonn
• tabl

Comments

Or as a square array read by antidiagonals, T(n,k) = Sum_{j=0..n} binomial(k,j)*binomial(n+k-j,k)*((j+1) mod 2).

A symmetric number triangle based on 1/(1-x^2).

The construction of a symmetric triangle in this example is general. Let f(n) be a sequence, preferably with f(0)=1. Then T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*f(j) yields a symmetric triangle. When f(n)=1^n, we get Pascal's triangle. When f(n)=2^n, we get the Delannoy triangle (see A008288). In general, f(n)=k^n yields a (1,k,1)-Pascal triangle (see A081577, A081578). Row sums of triangle are A100131. Diagonal sums of the triangle are A108351. Triangle mod 2 is A106465.

Examples

			Triangle rows begin
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 21, 32, 21,  6,  1;
As a square array read by antidiagonals, rows begin
  1, 1,  1,   1,   1,    1,    1, ...
  1, 2,  3,   4,   5,    6,    7, ...
  1, 3,  7,  13,  21,   31,   43, ...
  1, 4, 13,  32,  65,  116,  189, ...
  1, 5, 21,  65, 161,  341,  645, ...
  1, 6, 31, 116, 341,  842, 1827, ...
  1, 7, 43, 189, 645, 1827, 4495, ...
		

Programs

PARI

trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ Michel Marcus, Sep 11 2013

Formula

Row k (and column k) has g.f. (1+C(k,2)x^2)/(1-x)^(k+1).