A100226 Triangle, read by rows, of the coefficients of [x^k] in G100225(x)^n such that the row sums are 3^n-1 for n>0, where G100225(x) is the g.f. of A100225.

1, 1, 1, 1, 2, 5, 1, 3, 9, 13, 1, 4, 14, 28, 33, 1, 5, 20, 50, 85, 81, 1, 6, 27, 80, 171, 246, 197, 1, 7, 35, 119, 301, 553, 693, 477, 1, 8, 44, 168, 486, 1064, 1724, 1912, 1153, 1, 9, 54, 228, 738, 1854, 3600, 5220, 5193, 2785, 1, 10, 65, 300, 1070, 3012, 6730, 11760

Offset: 0

Author

Paul D. Hanna, Nov 28 2004

Keywords

• nonn
• tabl

Comments

Main diagonal forms A100227. Secondary diagonal is: T(n+1,n) = (n+1)*A001333(n), where A001333 is the numerators of continued fraction convergents to sqrt(2). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

Examples

			Rows begin:
  [1],
  [1,1],
  [1,2,5],
  [1,3,9,13],
  [1,4,14,28,33],
  [1,5,20,50,85,81],
  [1,6,27,80,171,246,197],
  [1,7,35,119,301,553,693,477],
  [1,8,44,168,486,1064,1724,1912,1153],...
where row sums form 3^n-1 for n>0:
3^1-1 = 1+1
3^2-1 = 1+2+5
3^3-1 = 1+3+9+13
3^4-1 = 1+4+14+28+33
3^5-1 = 1+5+20+50+85+81.
The main diagonal forms A100227 = [1,1,5,13,33,81,197,477,...], where Sum_{n>=1} (A100227(n)/n)*x^n = log((1-x)/(1-2*x-x^2)).
		

Crossrefs

Cf. A100225, A100227, A001333.

Programs

PARI

T(n,k,m=3)=if(n
				

Formula

G.f.: A(x, y)=(1-2*x*y+3*x^2*y^2)/((1-x*y)*(1-2*x*y-x^2*y^2-x*(1-x*y))).