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%I #48 Oct 08 2019 19:33:14
%S 1,1,10,150,2625,49875,997500,20662500,439078125,9513359375,
%T 209293906250,4661546093750,104884787109375,2380077861328125,
%U 54401779687500000,1251240932812500000,28934946571289062500
%N 5th-order Patalan numbers (generalization of Catalan numbers).
%H Vincenzo Librandi, <a href="/A025750/b025750.txt">Table of n, a(n) for n = 0..200</a>
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H T. M. Richardson, <a href="http://arxiv.org/abs/1410.5880">The Super Patalan Numbers</a>, arXiv preprint arXiv:1410.5880 [math.CO], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">J. Int. Seq. 18 (2015) # 15.3.3</a> .
%F G.f.: (6-(1-25*x)^(1/5))/5.
%F a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2; 4*A034301(n-1)= (5*n-6)(!^5) := product(5*j-6, j=2..n). - _Wolfdieter Lang_
%F a(n) = (sum(k=0..n-1, (-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(j=0..k, 2^j*binomial(k,j)*sum(i=j..n-k+j-1, binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)))))/n, n>0, a(0)=1. - _Vladimir Kruchinin_, Dec 10 2011
%F a(n) = ((-5)^(n-1)*sum(k=1..n, (5)^(n-k)*stirling1(n,k)))/n!, n>0, a(0)=1. - _Vladimir Kruchinin_, Mar 19 2013
%t CoefficientList[Series[(6-(1-25x)^(1/5))/5,{x,0,20}],x] (* _Harvey P. Dale_, Dec 06 2012 *)
%t a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Mar 19 2013, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i),i,j,n-k+j-1),j,0,k),k,0,n-1))/(n); /* _Vladimir Kruchinin_, Dec 10 2011 */
%o (Maxima)
%o a(n):=if n=0 then 1 else -binomial(1/5,n)*(-25)^n/5; /* _Tani Akinari_, Sep 17 2015 */
%Y Cf. A034687, A049393.
%K nonn
%O 0,3
%A _Olivier Gérard_
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