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

0,2

Comments

Matrix inverse is A104030. Antidiagonal sums form unsigned A078039.

Links

Table of n, a(n) for n=0..61.

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.

Example

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; ...

Prog

(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)}

Crossrefs

Cf. A104030, A104027, A078039.

Sequence in context: A210225 A180906 A153277 * A208752 A119308 A110197

Adjacent sequences: A104026 A104027 A104028 * A104030 A104031 A104032

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 26 2005

Status

approved