A185675 Riordan array ( (1+x)/(1-x-x^2), x*A000108(x) ).

1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 8, 17, 12, 5, 1, 13, 43, 35, 18, 6, 1, 21, 116, 103, 60, 25, 7, 1, 34, 333, 312, 196, 93, 33, 8, 1, 55, 1010, 976, 643, 331, 135, 42, 9, 1, 89, 3202, 3147, 2137, 1161, 518, 187, 52, 10, 1, 144, 10504, 10415, 7213, 4066, 1929, 768, 250, 63, 11, 1

Offset: 0

Author

Vladimir Kruchinin, Feb 09 2011

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   1;
   2,   1;
   3,   3,   1;
   5,   7,   4,   1;
   8,  17,  12,   5,   1;
  13,  43,  35,  18,   6,   1;
  21, 116, 103,  60,  25,   7,   1;
  34, 333, 312, 196,  93,  33,   8,   1;
Production matrix begins:
    2, 1;
   -1, 1, 1;
    2, 1, 1, 1;
   -3, 1, 1, 1, 1;
    5, 1, 1, 1, 1, 1;
   -8, 1, 1, 1, 1, 1, 1;
   13, 1, 1, 1, 1, 1, 1, 1;
  -21, 1, 1, 1, 1, 1, 1, 1, 1;
  ... _Philippe Deléham_, Sep 21 2014
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.

Programs

Maple

A185675 := proc(n,k) if n = k then 1; elif k = 0 then combinat[fibonacci](n+2) ; else k*add(1/(n-i)*add(binomial(j,i+1-j)*binomial(2*n-2*i-k-1,n-i-1), j=1..i+1), i=0..n-k) ; end if; end proc:
seq(seq(A185675(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 10 2011

Mathematica

r[n_, k_] := k*Sum[Binomial[2*(n - i) - k - 1, n - i - 1]*Fibonacci[i + 2]/(n - i), {i, 0, n - k}]; r[n_, 0] := Fibonacci[n + 2]; r[n_, n_] := 1; Table[r[n, k], {n, 0, 3}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013; modified by G. C. Greubel, Jul 10 2017 *)

Formula

R(n,k) = k*Sum_{i=0..n-k}(Sum_{j=1..i+1}binomial(j,i+1-j))*binomial(2*(n-i)-k-1,n-i-1)/(n-i), k>0.

R(n,0) = A000045(n+2).