OFFSET
0,2
COMMENTS
Partial column sums triangle of odd-indexed Fibonacci numbers.
Left border = odd-indexed Fibonacci numbers, next-to-left border = even-indexed Fibonacci numbers. Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).
Diagonal sums are A027994(n). - Philippe Deléham, Jan 14 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..568
FORMULA
Let the left border = odd-indexed Fibonacci numbers, (1, 2, 5, 13, 34...); then for k>1, T(n,k) = T(n-1,k) + T(n-1,k-1).
G.f.: (1-x)^2/((1-3*x+x^2)*(1-x*(1+y))). - Paul Barry, Dec 05 2006
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - 3*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(2,0)=5, T(2,1)=3, T(2,2)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 14 2014
Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(13 + 8*x + 4*x^2/2! + x^3/3!) = 13 + 21*x + 33*x^2/2! + 50*x^3/3! + 73*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
T(n,k) = C(n, n-k) + Sum_{i = 1..n} Fibonacci(2*i)*C(n-i, n-k-i), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. - Peter Bala, Mar 21 2018
EXAMPLE
(6,3) = 33 = 12 + 21 = (5,3) + (5,2). First few rows of the triangle are:
1;
2, 1;
5, 3, 1;
13, 8, 4, 1;
34, 21, 12, 5, 1;
89, 55, 33, 17, 6, 1;
...
MAPLE
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
end proc:
with(combinat):
for n from 0 to 10 do
seq(C(n, n-k) + add(fibonacci(2*i)*C(n-i, n-k-i), i = 1..n), k = 0..n);
end do; # Peter Bala, Mar 21 2018
PROG
(PARI)
T(n, k)=if(k==n, 1, if(k<=1, fibonacci(2*n-1), T(n-1, k)+T(n-1, k-1)));
for(n=1, 15, for(k=1, n, print1(T(n, k), ", ")); print()); /* show triangle */
/* Joerg Arndt, Jun 17 2011 */
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Nov 22 2006
EXTENSIONS
New description from Paul Barry, Dec 05 2006
Data error corrected by Johannes W. Meijer, Jun 16 2011
STATUS
approved