A099603 -id:A099603 - cp's OEIS frontend

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

Paul D. Hanna, Oct 25 2004

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).

			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,...].

Cf. A027907, A082759, A099603, A099604.

Cf. A104495.

{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)}

{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]}

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

A099606 Row sums of triangle A099605, in which row n equals the inverse Binomial transform of column n of the triangle A034870 of even-indexed rows of Pascal's triangle.

Original entry on oeis.org

1, 4, 10, 48, 116, 560, 1352, 6528, 15760, 76096, 183712, 887040, 2141504, 10340096, 24963200, 120532992, 290992384, 1405035520, 3392055808, 16378294272, 39540700160, 190919389184, 460920178688, 2225519493120, 5372879343616

Offset: 0

Paul D. Hanna, Oct 25 2004

nonn

			Sequence begins: {1*1, 2*2, 5*2, 12*4, 29*4, 70*8, 169*8, 408*16, ...}.

Index entries for linear recurrences with constant coefficients, signature (0, 12, 0, -4).

Cf. A099603, A099605, A034870, A000129.

a(n)=polcoeff((1+4*x-2*x^2)/(1-12*x^2+4*x^4)+x*O(x^n),n)

a(n) = Pell(n+1)*2^[(n+1)/2]. a(n) = 12*a(n-2) - 4*a(n-4) for n>=4. G.f.: (1+4*x-2*x^2)/(1-12*x^2+4*x^4).

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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.

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A099606 Row sums of triangle A099605, in which row n equals the inverse Binomial transform of column n of the triangle A034870 of even-indexed rows of Pascal's triangle.

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