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A185815 Exponential Riordan array (log(1/(1-x)), x*A005043(x)).

%I A185815 #32 Apr 10 2025 06:50:37
%S A185815 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 A185815 15,6,0,720,41580,22344,3990,910,21,7,0,5040,789984,379680,98784,
%U A185815 10500,1792,28,8,0,40320,16961616,8595936,1930320,325584,23436,3192,36,9,0
%N A185815 Exponential Riordan array (log(1/(1-x)), x*A005043(x)).
%H A185815 G. C. Greubel, <a href="/A185815/b185815.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185815 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 A185815 R(n,k):= (n!/(k-1)!)*Sum_{i=1..(n-k)} (1/(i*(n-i)))*Sum_{j=k..(n-i)} binomial(2*j-k-1,j-1)*(-1)^(n-j-i)*binomial(n-i,j), k>0, R(0,0)=0, R(n,0)=(n-1)!.
%e A185815 Array begins:
%e A185815     0;
%e A185815     1,     0;
%e A185815     1,     2,     0;
%e A185815     2,     3,     3,    0;
%e A185815     6,    32,     6,    4,   0;
%e A185815    24,   210,   140,   10,   5,  0;
%e A185815   120,  2904,  1170,  400,  15,  6, 0;
%e A185815   720, 41580, 22344, 3990, 910, 21, 7, 0;
%p A185815 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 A185815 seq(seq(A185815(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Feb 09 2011
%t A185815 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 A185815 nonn,tabl
%O A185815 0,5
%A A185815 _Vladimir Kruchinin_, Feb 05 2011
%E A185815 More terms from _Michel Marcus_, Feb 19 2025