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Exponential Riordan array (log(1/(1-x)), x*A005043(x)).
1

%I #23 Jul 15 2017 02:09:42

%S 0,1,0,1,2,0,2,3,3,0,6,32,6,4,0,24,210,140,10,5,0,120,2904,1170,400,

%T 15,6,0,720,41580,22344,3990,910,21,7,0,5040,789984,379680,98784,

%U 10500,1792,28,8,0,40320,16961616,8595936

%N Exponential Riordan array (log(1/(1-x)), x*A005043(x)).

%H G. C. Greubel, <a href="/A185815/b185815.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):= (n!/(k-1))*Sum_{i=1..(n-k)} (1/i)*Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j))/(n-i), k>0, R(0,0)=0, R(n,0)=(n-1)!.

%e Array begins:

%e 0;

%e 1, 0;

%e 1, 2, 0;

%e 2, 3, 3, 0;

%e 6, 32, 6, 4, 0;

%e 24, 210, 140, 10, 5, 0;

%e 120, 2904, 1170, 400, 15, 6, 0;

%e 720, 41580, 22344, 3990, 910, 21, 7, 0;

%p A185815 := proc(n,k) if n = k then 0; elif k = 0 then (n-1)! ; else n!/(k-1)!*add(1/i/(n-i)*add(binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j),j=k..n-i),i=1..n-k) ; end if; end proc:

%p seq(seq(A185815(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 09 2011

%t t[n_, k_] := n!/(k-1)!*Sum[ 1/(i*(n-i))*((-1)^(n+k-i)*(n-i)!*HypergeometricPFQ[ {(k+1)/2, k/2, i+k-n}, {k, k+1}, 4]) / (k!*(n-k-i)!), {i, 1, n-k}]; t[0, 0] = 0; t[n_, 0] := (n-1)!; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 01 2013, after given formula *)

%K nonn,tabl

%O 0,5

%A _Vladimir Kruchinin_, Feb 05 2011