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A104029 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified August 1 17:43 EDT 2021. Contains 346402 sequences. (Running on oeis4.)