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A185814 Exponential Riordan array (e^x,A005043(x)).

%I A185814 #21 Jun 02 2025 03:54:57
%S A185814 1,1,1,1,2,1,1,9,3,1,1,52,30,4,1,1,545,250,70,5,1,1,6966,3615,740,135,
%T A185814 6,1,1,114457,56301,13895,1715,231,7,1,1,2199464,1107148,255416,40390,
%U A185814 3416,364,8,1,1,49219137,24542820,5904444,856926,98406,6132,540,9,1
%N A185814 Exponential Riordan array (e^x,A005043(x)).
%H A185814 G. C. Greubel, <a href="/A185814/b185814.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A185814 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 A185814 R(n,k) = (n!/(k-1)!)*Sum_{i=0..(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(n,0)=1.
%e A185814 [1]
%e A185814 [1,1]
%e A185814 [1,2,1]
%e A185814 [1,9,3,1]
%e A185814 [1,52,30,4,1]
%e A185814 [1,545,250,70,5,1]
%e A185814 [1,6966,3615,740,135,6,1]
%e A185814 [1,114457,56301,13895,1715,231,7,1]
%t A185814 r[n_, 0] := 1; r[n_, k_] := (n!/(k - 1)!)*Sum[(1/i!)*Sum[Binomial[2*j - k - 1, j - 1]*(-1)^(n - j - i)*Binomial[n - i, j], {j, k, n - i}]/(n - i), {i, 0, n - k}]; Table[r[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Jul 14 2017 *)
%K A185814 nonn,tabl
%O A185814 0,5
%A A185814 _Vladimir Kruchinin_, Feb 05 2011