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Riordan array (A000045(x), x*A005043(x)).
1

%I #27 Jul 15 2017 02:22:11

%S 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,

%T 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,

%U 152,82,45,15,9,1,1,0

%N Riordan array (A000045(x), x*A005043(x)).

%H G. C. Greubel, <a href="/A185813/b185813.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2013.

%F 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.

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

%e Array begins:

%e 0;

%e 1, 0;

%e 1, 1, 0;

%e 2, 1, 1, 0;

%e 3, 3, 1, 1, 0;

%e 5, 5, 4, 1, 1, 0;

%e 8, 11, 7, 5, 1, 1, 0;

%e 13, 22, 18, 9, 6, 1, 1, 0;

%e 21, 48, 39, 26, 11, 7, 1, 1, 0;

%e 34, 106, 94, 59, 35, 13, 8, 1, 1, 0;

%e 55, 245, 223, 152, 82, 45, 15, 9, 1, 1, 0;

%p 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:

%p seq(seq(A185813(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 10 2011

%t 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 *)

%Y Cf. A000045 (Fibonacci).

%K nonn,tabl

%O 0,7

%A _Vladimir Kruchinin_, Feb 05 2011