A099605 - OEIS

%I #8 Apr 15 2017 00:54:43

%S 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,

%T 456,272,64,8,112,560,1408,1992,1616,704,128,1,44,406,1652,3649,4712,

%U 3568,1472,256,10,210,1572,6084,14002,20330,18880,10912,3584,512,1,65

%N 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.

%C 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.

%H G. C. Greubel, <a href="/A099605/b099605.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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.

%e Rows begin:

%e [1],

%e [2,2],

%e [1,5,4],

%e [4,16,20,8],

%e [1,14,41,44,16],

%e [6,50,146,198,128,32],

%e [1,27,155,377,456,272,64],

%e [8,112,560,1408,1992,1616,704,128],

%e [1,44,406,1652,3649,4712,3568,1472,256],

%e [10,210,1572,6084,14002,20330,18880,10912,3584,512],

%e [1,65,870,5202,17469,36365,48940,42800,23552,7424,1024],...

%e The binomial transform of row 2 equals column 2 of A034870:

%e BINOMIAL[1,5,4] = [1,6,15,28,45,66,91,120,153,...].

%e The binomial transform of row 3 equals column 3 of A034870:

%e BINOMIAL[4,16,20,8] = [4,20,56,120,220,364,560,...].

%e The binomial transform of row 4 equals column 4 of A034870:

%e BINOMIAL[1,14,41,44,16] = [1,15,70,210,495,1001,...].

%t 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],

%t   y] // Flatten (* _G. C. Greubel_, Apr 14 2017 *)

%o (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)}

%Y Cf. A099602, A099605, A034870, A000129.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Oct 25 2004