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
Examples
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,...].
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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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 *) -
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)}
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.
Comments