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%I #33 Dec 11 2019 09:47:53
%S 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,
%T 1,610,377,232,138,73,30,8,1,1597,987,609,370,211,103,38,9,1,4181,
%U 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
%N Riordan array ((1-x)/(1-3*x+x^2),x/(1-x)) read by rows.
%C Partial column sums triangle of odd-indexed Fibonacci numbers.
%C Left border = odd-indexed Fibonacci numbers, next-to-left border = even-indexed Fibonacci numbers. Row sums = A061667: (1, 3, 9, 26, 73, 201, ...).
%C Diagonal sums are A027994(n). - _Philippe Deléham_, Jan 14 2014
%H Vincenzo Librandi, <a href="/A125171/b125171.txt">Table of n, a(n) for n = 0..568</a>
%H Peter Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%F 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).
%F G.f.: (1-x)^2/((1-3*x+x^2)*(1-x*(1+y)). - _Paul Barry_, Dec 05 2006
%F 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
%F 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
%F 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
%e (6,3) = 33 = 12 + 21 = (5,3) + (5,2). First few rows of the triangle are:
%e 1;
%e 2, 1;
%e 5, 3, 1;
%e 13, 8, 4, 1;
%e 34, 21, 12, 5, 1;
%e 89, 55, 33, 17, 6, 1;
%e ...
%p C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if;
%p end proc:
%p with(combinat):
%p for n from 0 to 10 do
%p seq(C(n, n-k) + add(fibonacci(2*i)*C(n-i, n-k-i), i = 1..n), k = 0..n);
%p end do; # _Peter Bala_, Mar 21 2018
%o (PARI)
%o T(n,k)=if(k==n,1,if(k<=1,fibonacci(2*n-1),T(n-1,k)+T(n-1,k-1)));
%o for(n=1,15,for(k=1,n,print1(T(n,k),", "));print()); /* show triangle */
%o /* _Joerg Arndt_, Jun 17 2011 */
%Y Cf. A027994, A061667 (row sums).
%K nonn,tabl,easy
%O 0,2
%A _Gary W. Adamson_, Nov 22 2006
%E New description from _Paul Barry_, Dec 05 2006
%E Data error corrected by _Johannes W. Meijer_, Jun 16 2011
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