A100224 Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.

1, 1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 4, 6, 1, 0, 5, 5, 10, 10, 1, 0, 6, 6, 15, 18, 17, 1, 0, 7, 7, 21, 28, 35, 28, 1, 0, 8, 8, 28, 40, 60, 64, 46, 1, 0, 9, 9, 36, 54, 93, 117, 117, 75, 1, 0, 10, 10, 45, 70, 135, 190, 230, 210, 122

Offset: 0

Author

Paul D. Hanna, Nov 28 2004

Keywords

• nonn
• tabl

Comments

Diagonals are: T(n,n)=A001610(n-1) for n>0, with T(0,0)=1, T(n+1,n)=A006490(n), T(n+2,n)=A006491(n), T(n+3,n)=A006492(n), T(n+4,n)=A006493(n). The ratio of the generating functions of any two adjacent diagonals gives: (1-x)/(1-x-x^2) = 1+ x^2+ x^3+ 2*x^4+ 3*x^5+ 5*x^6+ 8*x^7+ 13*x^8+...

Examples

			Rows begin:
  [1],
  [1,0],
  [1,0,2],
  [1,0,3,3],
  [1,0,4,4,6],
  [1,0,5,5,10,10],
  [1,0,6,6,15,18,17],
  [1,0,7,7,21,28,35,28],
  [1,0,8,8,28,40,60,64,46],...
where row sums form 2^n-1 for n>0:
2^1-1 = 1+0 = 1
2^2-1 = 1+0+2 = 3
2^3-1 = 1+0+3+3 = 7
2^4-1 = 1+0+4+4+6 = 15
2^5-1 = 1+0+5+5+10+10 = 31.
The main diagonal forms A001610 = [0,2,3,6,10,17,...], where Sum_{n>=1} (A001610(n-1)/n)*x^n = log((1-x)/(1-x-x^2)).
		

Crossrefs

Cf. A100223, A001610, A006490, A006491, A006492, A006493.

Programs

PARI

T(n,k)=if(n
				

Formula

G.f.: A(x, y)=(1-2*x*y+2*x^2*y^2)/((1-x*y)*(1-x*y-x^2*y^2-x*(1-x*y))).