0,5

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))).

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))).

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).

(PARI) {T(n, k, m=3)=if(n<k|k<0, 0, if(k==0, 1, polcoeff(((1+(m-1)*x+sqrt(1+2*(m-3)*x+(m^2-2*m+5)*x^2+x*O(x^k)))/2)^n, k)))}

Cf. A100225, A100227, A001333.

Sequence in context: A091953 A011456 A100084 this_sequence A121428 A105686 A153726

Adjacent sequences: A100223 A100224 A100225 this_sequence A100227 A100228 A100229

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2004