OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
FORMULA
G.f.: 5 / (4+(1-25*x)^(1/5)).
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
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
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
MAPLE
# Based on Tani Akinari's formula.
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):
seq(round(evalf(a(n), 64)), n=0..16); # Peter Luschny, Sep 21 2015
MATHEMATICA
Table[SeriesCoefficient[5/(4 + (1 - 25*x)^(1/5)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved