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

0,6

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+...

Links

Table of n, a(n) for n=0..65.

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))).

Example

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).

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, polcoeff(((1+x+sqrt(1-2*x+5*x^2+x*O(x^k)))/2)^n, k)))

Crossrefs

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

Sequence in context: A122286 A122202 A122285 * A208671 A208727 A242378

Adjacent sequences: A100221 A100222 A100223 * A100225 A100226 A100227

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 28 2004

Status

approved