cp's OEIS Frontend

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

%I A185813 #37 Feb 19 2025 10:25:11
%S A185813 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 A185813 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 A185813 152,82,45,15,9,1,1,0
%N A185813 Riordan array (A000045(x), x*A005043(x)).
%H A185813 G. C. Greubel, <a href="/A185813/b185813.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185813 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A185813 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 A185813 R(n,0) = Fibonacci(n).
%e A185813 Array begins:
%e A185813    0;
%e A185813    1,   0;
%e A185813    1,   1,   0;
%e A185813    2,   1,   1,   0;
%e A185813    3,   3,   1,   1,  0;
%e A185813    5,   5,   4,   1,  1,  0;
%e A185813    8,  11,   7,   5,  1,  1,  0;
%e A185813   13,  22,  18,   9,  6,  1,  1, 0;
%e A185813   21,  48,  39,  26, 11,  7,  1, 1, 0;
%e A185813   34, 106,  94,  59, 35, 13,  8, 1, 1, 0;
%e A185813   55, 245, 223, 152, 82, 45, 15, 9, 1, 1, 0;
%p A185813 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 A185813 seq(seq(A185813(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 10 2011
%t A185813 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 A185813 Cf. A000045 (Fibonacci).
%K A185813 nonn,tabl
%O A185813 0,7
%A A185813 _Vladimir Kruchinin_, Feb 05 2011