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A235340
10*binomial(11*n+10,n)/(11*n+10).
5
1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 242182964452000, 6127720969229265, 156431295179478200, 4024231652469275640, 104218796026870015374, 2714941275486017847825
OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=10.
LINKS
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
FORMULA
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=11, r=10.
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(11*n + 9, n + 1)/(10*n + 9) which is instance k = 10 of c(k, n+1) given in a comment in A130564.
x*B(x), with the g.f. above named B(x), is the compositional inverse of y*(1 - y)^10, hence B(x)*(1 - x*B(x))^10 = 1.
G.f.: 11F10([10..20]/11, [11..20]/10; (11^11/10^10)*x) = (10/(11*x))*(1 - 10F9([-1,1,2,3,4,5,6,7,8,9]/11, [1,2,3,4,5,6,7,8,9]/10; (11^11/10^10)*x)).
(End)
MATHEMATICA
Table[10 Binomial[11 n + 10, n]/(11 n + 10), {n, 0, 30}]
PROG
(PARI) a(n) = 10*binomial(11*n+10, n)/(11*n+10);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/10))^10+x*O(x^n)); polcoeff(B, n)}
(Magma) [10*Binomial(11*n+10, n)/(11*n+10): n in [0..30]];
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Jan 06 2014
STATUS
approved