A067418 - OEIS

%I #17 Apr 11 2016 23:13:21

%S 1,2,1,5,3,2,10,7,5,3,20,15,12,8,5,38,30,25,19,13,8,71,58,50,40,31,21,

%T 13,130,109,96,80,65,50,34,21,235,201,180,154,130,105,81,55,34,420,

%U 365,331,289,250,210,170,131,89,55,744,655,600,532,469,404,340,275,212,144,89,1308,1164,1075,965

%N Triangle A067330 with rows read backwards.

%C The column m (without leading 0's) gives the convolution of Fibonacci numbers F(n+1) := A000045(n+1), n>=0, with those with m-shifted index: a(n+m,m)=sum(F(k+1)*F(m+n+1-k),k=0..n), n>=0, m=0,1,...

%C The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. for Fibonacci F(n+1), n>=0).

%C The columns give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for m= 0..9, respectively. Row sums give A067988.

%H Michael De Vlieger, <a href="/A067418/b067418.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n, m)=A067330(n, n-m), n>=m>=0, else 0.

%F a(n, m)= (((3*(n-m)+5)*F(n-m+1)+(n-m+1)*F(n-m))*F(m+1)+((n-m)*F(n-m+1)+2*(n-m+1)*F(n-m))*F(m))/5.

%F G.f. for column m=0, 1, ...: (x^m)*(F(m+1)+F(m)*x)/(1-x-x^2)^2, with F(m) := A000045(m) (Fibonacci).

%F a(n, m) = ((-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(n-m+1)*L(n+2)-A067990(n, m))/5, n>=m>=0. - _Ehren Metcalfe_, Apr 11 2016

%e {1}; {2,1}; {5,3,2}; {10,7,5,3}; ...; p(2,n)=5+3*x+2*x^2.

%t Reverse /@ Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 11 2016 *)

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Feb 15 2002