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A025758 5th-order Vatalan numbers (generalization of Catalan numbers).

%I A025758 #39 Aug 04 2024 12:44:03
%S A025758 1,1,11,171,3056,58916,1191376,24896436,532911346,11617952106,
%T A025758 256966100966,5750337968926,129926216608236,2959472057112396,
%U A025758 67877180959091156,1566072624078270516,36319953436423545851
%N A025758 5th-order Vatalan numbers (generalization of Catalan numbers).
%H A025758 Vincenzo Librandi, <a href="/A025758/b025758.txt">Table of n, a(n) for n = 0..200</a>
%H A025758 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 A025758 G.f.: 5 / (4+(1-25*x)^(1/5)).
%F A025758 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 A025758 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
%F A025758 a(n) = (-1)^(n+1) * 5^(2*n+1) * Sum_{k>=0} (-1/4)^(k+1) * binomial(k/5,n). - _Seiichi Manyama_, Aug 04 2024
%p A025758 # Based on Tani Akinari's formula.
%p A025758 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 A025758 seq(round(evalf(a(n),64)),n=0..16); # _Peter Luschny_, Sep 21 2015
%t A025758 Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Dec 29 2012 *)
%o A025758 (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 A025758 a(n), n >= 1, = row sums of triangle A049223.
%Y A025758 Cf. A000108, A025756, A025757, A025759, A025760, A025761, A025762, A025763.
%K A025758 nonn
%O A025758 0,3
%A A025758 _Olivier Gérard_