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
Examples
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,...].
Programs
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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]}
Formula
G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4).
Comments