A125171 Riordan array ((1-x)/(1-3*x+x^2),x/(1-x)) read by rows.
1, 2, 1, 5, 3, 1, 13, 8, 4, 1, 34, 21, 12, 5, 1, 89, 55, 33, 17, 6, 1, 233, 144, 88, 50, 23, 7, 1, 610, 377, 232, 138, 73, 30, 8, 1, 1597, 987, 609, 370, 211, 103, 38, 9, 1, 4181, 2584, 1596, 979, 581, 314, 141, 47, 10, 1, 10946, 6765, 4180, 2575, 1560, 895, 455, 188, 57, 11, 1, 28657, 17711, 10945, 6755, 4135, 2455, 1350, 643
Offset: 0
Examples
(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; ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..568
- Peter Bala, A note on the diagonals of a proper Riordan Array
Programs
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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 -
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 */
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
Extensions
New description from Paul Barry, Dec 05 2006
Data error corrected by Johannes W. Meijer, Jun 16 2011
Comments