A092422 Triangle, read by rows, where T(n,k) equals the k-th term of the convolution of the (n-k)-th row with the (2k)-th Fibonacci polynomial (A011973).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 7, 1, 1, 5, 14, 18, 11, 1, 1, 6, 21, 40, 36, 16, 1, 1, 7, 30, 72, 98, 66, 22, 1, 1, 8, 40, 119, 211, 214, 113, 29, 1, 1, 9, 52, 182, 398, 546, 428, 183, 37, 1, 1, 10, 65, 265, 692, 1170, 1278, 799, 283, 46, 1, 1, 11, 80, 368, 1123, 2286, 3104

Offset

0,5

Links

Table of n, a(n) for n=0..72.

Formula

T(n, k) = sum_{j=0, min(k, n-k)} binomial(k+j, k-j)*T(n-k, j) with T(n, 0)=1.

Example

Even-numbered Fibonacci polynomials (cf. A011973) are:
{1},
{1,1},
{1,3,1},
{1,5,6,1},
{1,7,15,10,1},...
These terms are used to generate each row from the prior rows. For example,
row 5 = {1(1), 1(1)+1(4), 1(1)+3(3)+1(4), 1(1)+6(2)+5(1), 1(1)+10(1), 1(1)};
row 6 = {1(1), 1(1)+1(5), 1(1)+3(4)+1(8), 1(1)+6(3)+5(4)+1(1), 1(1)+10(2)+15(1), 1(1)+15(1), 1(1)}.
Rows begin:
{1},
{1,1},
{1,2,1},
{1,3,4,1},
{1,4,8,7,1},
{1,5,14,18,11,1},
{1,6,21,40,36,16,1},
{1,7,30,72,98,66,22,1},
{1,8,40,119,211,214,113,29,1},
{1,9,52,182,398,546,428,183,37,1},...

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, sum(j=0, min(k, n-k), binomial(k+j, k-j)*T(n-k, j))))

Crossrefs

Cf. A092423, A092424, A011973.

Sequence in context: A258309 A197957 A089899 * A096465 A124460 A144042

Adjacent sequences: A092419 A092420 A092421 * A092423 A092424 A092425

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 22 2004

Status

approved