A099605 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle A034870 of coefficients in successive powers of the trinomial (1+2*x+x^2), omitting leading zeros.

1, 2, 2, 1, 5, 4, 4, 16, 20, 8, 1, 14, 41, 44, 16, 6, 50, 146, 198, 128, 32, 1, 27, 155, 377, 456, 272, 64, 8, 112, 560, 1408, 1992, 1616, 704, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 10, 210, 1572, 6084, 14002, 20330, 18880, 10912, 3584, 512, 1, 65

Offset

0,2

Comments

Row sums form A099606, where A099606(n) = Pell(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A026000, where A026000(n) = T(2n,n), where T = Delannoy triangle (A008288). Antidiagonal sums form A099607.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

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

Example

Rows begin:
[1],
[2,2],
[1,5,4],
[4,16,20,8],
[1,14,41,44,16],
[6,50,146,198,128,32],
[1,27,155,377,456,272,64],
[8,112,560,1408,1992,1616,704,128],
[1,44,406,1652,3649,4712,3568,1472,256],
[10,210,1572,6084,14002,20330,18880,10912,3584,512],
[1,65,870,5202,17469,36365,48940,42800,23552,7424,1024],...
The binomial transform of row 2 equals column 2 of A034870:
BINOMIAL[1,5,4] = [1,6,15,28,45,66,91,120,153,...].
The binomial transform of row 3 equals column 3 of A034870:
BINOMIAL[4,16,20,8] = [4,20,56,120,220,364,560,...].
The binomial transform of row 4 equals column 4 of A034870:
BINOMIAL[1,14,41,44,16] = [1,15,70,210,495,1001,...].

Mathematica

CoefficientList[CoefficientList[Series[(1 + 2*(y + 1)*x - (y + 1)*x^2)/(1 - (2*y + 1)*(2*y + 2)*x^2 + (y + 1)^2*x^4), {x, 0, 49}, {y, 0, 49}], x],
  y] // Flatten (* G. C. Greubel, Apr 14 2017 *)

Prog

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

Crossrefs

Cf. A099602, A099605, A034870, A000129.

Sequence in context: A125178 A101975 A136388 * A288421 A079218 A079220

Adjacent sequences: A099602 A099603 A099604 * A099606 A099607 A099608

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 25 2004

Status

approved