A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).
1, 2, 1, 3, 5, 1, 4, 13, 9, 1, 5, 26, 35, 14, 1, 6, 45, 96, 75, 20, 1, 7, 71, 216, 267, 140, 27, 1, 8, 105, 427, 750, 623, 238, 35, 1, 9, 148, 770, 1800, 2123, 1288, 378, 44, 1, 10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1, 11, 265, 2067, 7590, 15115, 17303, 11505
Offset: 0
Examples
Row 3: {4,13,9,1} is formed from the pairwise sums
of row 3 of A027907: {1,3, 6,7, 6,3, 1}.
Rows begin:
1;
2, 1;
3, 5, 1;
4, 13, 9, 1;
5, 26, 35, 14, 1;
6, 45, 96, 75, 20, 1;
7, 71, 216, 267, 140, 27, 1;
8, 105, 427, 750, 623, 238, 35, 1;
9, 148, 770, 1800, 2123, 1288, 378, 44, 1;
10, 201, 1296, 3858, 6046, 5211, 2436, 570, 54, 1;
11, 265, 2067, 7590, 15115, 17303, 11505, 4302, 825, 65, 1;
12, 341, 3157, 13959, 34210, 49721, 43923, 23397, 7194, 1155, 77, 1; ...
Programs
-
PARI
{T(n,k)=polcoeff((1+x+x^2)^n+x*O(x^(2*k)),2*k)+ polcoeff((1+x+x^2)^n+x*O(x^(2*k+1)),2*k+1)} for(n=0,10,for(k=0,n,print1(T(n,k),", "));print("")) -
PARI
{T(n,k)=polcoeff(polcoeff((1-x*y)/(1-2*x*(1+y)+x^2*(1+y+y^2)) +x*O(x^n),n,x)+y*O(y^k),k,y)}
Formula
G.f.: A(x, y) = (1-x*y)/(1 - 2*x*(1+y) + x^2*(1+y+y^2) ).
T(n, k) = [x^(2k)](1+x+x^2)^n + [x^(2k+1)](1+x+x^2)^n.
Comments