0,5

Row sums form A099603, where A099603(n) = fibonacci(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A082759, where A082759(n) = Sum_{k=0..n} binomial(n,k)*Trinomial(n,k). Antidiagonal sums form A099604.

Matrix inverse equals triangle A104495, which is generated from self-convolutions of the Catalan sequence (A000108).

G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4).

Rows begin:

[1],

[1,1],

[1,2,1],

[2,5,4,1],

[1,5,8,5,1],

[3,13,22,18,7,1],

[1,9,26,35,24,8,1],

[4,26,70,101,84,40,10,1],

[1,14,61,131,160,116,49,11,1],

[5,45,171,363,476,400,215,71,13,1],

[1,20,120,363,654,752,565,275,83,14,1],...

The binomial transform of row 2 equals column 2 of A027907:

BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...].

The binomial transform of row 3 equals column 3 of A027907:

BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...].

The binomial transform of row 4 equals column 4 of A027907:

BINOMIAL[1,5,8,5,1] = [1,6,19,45,90,161,266,414,615,...].

The binomial transform of row 5 equals column 5 of A027907:

BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...].

(PARI) {T(n, k)=polcoeff(polcoeff((1+(y+1)*x-(y+1)*x^2)/(1-(y+1)*(y+2)*x^2+(y+1)^2*x^4)+x*O(x^n), n, x)+y*O(y^k), k, y)}

(PARI) {T(n, k)=(matrix(n+1, n+1, i, j, if(i>=j, polcoeff(polcoeff( (1+x*y/(1+x))/(1+x-y^2*(1-(1+4*x+O(x^i))^(1/2))^2/4+O(y^j)), i-1, x), j-1, y)))^-1)[n+1, k+1]}

Cf. A027907, A082759, A099603, A099604.

Cf. A104495.

Sequence in context: A124218 A025165 A106480 this_sequence A104560 A121435 A137156

Adjacent sequences: A099599 A099600 A099601 this_sequence A099603 A099604 A099605

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 25 2004