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%I #35 Sep 21 2015 07:48:19
%S 1,1,11,171,3056,58916,1191376,24896436,532911346,11617952106,
%T 256966100966,5750337968926,129926216608236,2959472057112396,
%U 67877180959091156,1566072624078270516,36319953436423545851
%N 5th-order Vatalan numbers (generalization of Catalan numbers).
%H Vincenzo Librandi, <a href="/A025758/b025758.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.
%F G.f.: 5 / (4+(1-25*x)^(1/5)).
%F a(n) = sum(m=1..n-1, 5^(n-m)*m/n * sum(k=1..n-m, binomial(n+k-1,n-1) * sum(i=0..k, binomial(k,i) * 2^(k-i) * sum(j=0..i, binomial(j,-3*i+n-m-k+2*j) * (-1)^(j-i)*5^(j-i)*(-2)^(3*i-n+m+k-j) * binomial(i,j)))))+1. - _Vladimir Kruchinin_, Feb 09 2011
%F Conjecture: 41*n*(n-1)*(n-2)*(n-3)*a(n) -3*(1367*n-4100)*(n-1)*(n-2)*(n-3)*a(n-1) +50*(n-2)*(n-3)*(3077*n^2-21533*n+38136)*a(n-2) -250*(n-3)*(10265*n^3-123135*n^2+494446*n-664572)*a(n-3) +1875*(8575*n^4-154250*n^3+1039765*n^2-3112730*n+3491808)*a(n-4) -625*(5*n-22)*(5*n-21)*(5*n-24)*(5*n-23)*a(n-5)=0. - _R. J. Mathar_, Jul 28 2014
%p # Based on Tani Akinari's formula.
%p h := (n,j) -> ((-1)^n/(-4)^j)*binomial(j/5,n+1)*hypergeom([1,n+1-j/5],[n+2], 1025): a := n -> 2^8*5^(2*n+1)*add(h(n,j), j=1..4):
%p seq(round(evalf(a(n),64)),n=0..16); # _Peter Luschny_, Sep 21 2015
%t Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Dec 29 2012 *)
%o (Maxima) a(n):=(256/205)*41^(-n)*sum(sum((-4)^(-k)*(-1025)^m*binomial(k/5,m),k,0,4),m,0,n); /* _Tani Akinari_, Sep 16 2015 */
%Y a(n), n >= 1, = row sums of triangle A049223.
%K nonn
%O 0,3
%A _Olivier Gérard_
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