OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
T. M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.3.3 .
FORMULA
G.f.: (6-(1-25*x)^(1/5))/5.
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
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
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
From Karol A. Penson, Feb 05 2025: (Start)
a(n) without the initial 1 (i.e. a(n) for n>=1) is given by
a(n+1) = 5^(2*n)*gamma(n + 4/5)/(gamma(4/5)*(n + 1)!),n>=0.
a(n+1) = Integral_{x=0..25} x^n*W(x), n>=0,
where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)
MATHEMATICA
CoefficientList[Series[(6-(1-25x)^(1/5))/5, {x, 0, 20}], x] (* Harvey P. Dale, Dec 06 2012 *)
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 *)
PROG
(Maxima)
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 */
(Maxima)
a(n):=if n=0 then 1 else -binomial(1/5, n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved