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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).
2

%I #6 Jun 13 2017 21:52:40

%S 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,

%T 30,72,98,66,22,1,1,8,40,119,211,214,113,29,1,1,9,52,182,398,546,428,

%U 183,37,1,1,10,65,265,692,1170,1278,799,283,46,1,1,11,80,368,1123,2286,3104

%N 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).

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

%e Even-numbered Fibonacci polynomials (cf. A011973) are:

%e {1},

%e {1,1},

%e {1,3,1},

%e {1,5,6,1},

%e {1,7,15,10,1},...

%e These terms are used to generate each row from the prior rows. For example,

%e 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)};

%e 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)}.

%e Rows begin:

%e {1},

%e {1,1},

%e {1,2,1},

%e {1,3,4,1},

%e {1,4,8,7,1},

%e {1,5,14,18,11,1},

%e {1,6,21,40,36,16,1},

%e {1,7,30,72,98,66,22,1},

%e {1,8,40,119,211,214,113,29,1},

%e {1,9,52,182,398,546,428,183,37,1},...

%o (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))))

%Y Cf. A092423, A092424, A011973.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 22 2004