A185813 Riordan array (A000045(x), x*A005043(x)).

0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 3, 1, 1, 0, 5, 5, 4, 1, 1, 0, 8, 11, 7, 5, 1, 1, 0, 13, 22, 18, 9, 6, 1, 1, 0, 21, 48, 39, 26, 11, 7, 1, 1, 0, 34, 106, 94, 59, 35, 13, 8, 1, 1, 0, 55, 245, 223, 152, 82, 45, 15, 9, 1, 1, 0

Offset: 0

Author

Vladimir Kruchinin, Feb 05 2011

Keywords

• nonn
• tabl

Examples

			Array begins:
   0;
   1,   0;
   1,   1,   0;
   2,   1,   1,   0;
   3,   3,   1,   1,  0;
   5,   5,   4,   1,  1,  0;
   8,  11,   7,   5,  1,  1,  0;
  13,  22,  18,   9,  6,  1,  1, 0;
  21,  48,  39,  26, 11,  7,  1, 1, 0;
  34, 106,  94,  59, 35, 13,  8, 1, 1, 0;
  55, 245, 223, 152, 82, 45, 15, 9, 1, 1, 0;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A000045 (Fibonacci).

Programs

Maple

A185813 := proc(n,k) if n = k then 0; elif k = 0 then combinat[fibonacci](n) ; else k*add(1/(n-i)*combinat[fibonacci](i)*add(binomial(2*j-k-1,j-1) *(-1)^(n-j-i) *binomial(n-i,j),j=k..n-i),i=0..n-k) ; end if; end proc:
seq(seq(A185813(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 10 2011

Mathematica

r[n_, k_] := k*Sum[((-1)^(n+k-i)*Fibonacci[i]*(n-i)!*HypergeometricPFQ[{k/2 + 1/2, k/2, i+k-n}, {k, k+1}, 4])/((n-i)*k!*(n-i-k)!), {i, 0, n-k}]; r[n_, 0] := Fibonacci[n]; Table[r[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

Formula

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

R(n,0) = Fibonacci(n).