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A125171 Riordan array ((1-x)/(1-3*x+x^2),x/(1-x)) read by rows. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

P. Bala, A note on the diagonals of a proper Riordan Array

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

Cf. A027994, A061667 (row sums).

Sequence in context: A054446 A164981 A047858 * A280784 A048472 A038622

Adjacent sequences:  A125168 A125169 A125170 * A125172 A125173 A125174

KEYWORD

nonn,tabl,easy

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

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Last modified April 23 03:26 EDT 2019. Contains 322380 sequences. (Running on oeis4.)