A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

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, 6, 71, 356, 1017, 1856, 2282, 1932

Offset

0,5

Comments

Row sums form A099603, where A099603(n) = Fibonacci(n+1)*2^floor((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).

Links

Table of n, a(n) for n=0..72.

Formula

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

Example

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 = column 2 of A027907: BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...].
The binomial transform of row 3 = column 3 of A027907: BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...].
The binomial transform of row 4 = 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 = column 5 of A027907: BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...].

Prog

(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]}

Crossrefs

Cf. A027907, A082759, A099603, A099604.

Cf. A104495.

Sequence in context: A318354 A348373 A106480 * A151703 A151691 A201780

Adjacent sequences: A099599 A099600 A099601 * A099603 A099604 A099605

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 25 2004

Status

approved