This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A194560 #34 Feb 16 2025 08:33:15
%S A194560 1,2,2,4,2,10,2,20,14,49,2,217,2,438,310,1580,2,6352,2,18062,7824,
%T A194560 58799,2,258971,2532,742915,246794,2729095,2,11154954,2,35779660,
%U A194560 8414818,129644809,242354,531132915,2,1767263211,300830821,6593815523,2,26289925026,2,91708135773
%N A194560 G.f.: Sum_{n>=1} G_n(x)^n where G_n(x) = x + x*G_n(x)^n.
%C A194560 Number of Dyck n-paths with all ascents of equal length. - _David Scambler_, Nov 17 2011
%C A194560 From _Gus Wiseman_, Feb 15 2019: (Start)
%C A194560 Also the number of uniform (all blocks have the same size) non-crossing set partitions of {1,...,n}. For example, the a(3) = 2 through a(6) = 10 uniform non-crossing set partitions are:
%C A194560 {{123}} {{1234}} {{12345}} {{123456}}
%C A194560 {{1}{2}{3}} {{12}{34}} {{1}{2}{3}{4}{5}} {{123}{456}}
%C A194560 {{14}{23}} {{126}{345}}
%C A194560 {{1}{2}{3}{4}} {{156}{234}}
%C A194560 {{12}{34}{56}}
%C A194560 {{12}{36}{45}}
%C A194560 {{14}{23}{56}}
%C A194560 {{16}{23}{45}}
%C A194560 {{16}{25}{34}}
%C A194560 {{1}{2}{3}{4}{5}{6}}
%C A194560 (End)
%H A194560 Paul D. Hanna, <a href="/A194560/b194560.txt">Table of n, a(n) for n = 1..1000</a>
%H A194560 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DyckPath.html">Dyck Path</a>.
%H A194560 Wikipedia, <a href="https://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>.
%F A194560 a(n) = Sum_{d|n} C(n,d)/(n-d+1).
%F A194560 G.f.: Sum_{n>=1} Series_Reversion( x/(1+x^n) )^n.
%e A194560 G.f.: A(x) = x + 2*x^2 + 2*x^3 + 4*x^4 + 2*x^5 + 10*x^6 + 2*x^7 + ...
%e A194560 where
%e A194560 A(x) = G_1(x) + G_2(x)^2 + G_3(x)^3 + G_4(x)^4 + G_5(x)^5 + ...
%e A194560 and G_n(x) = x + x*G_n(x)^n is given by:
%e A194560 G_n(x) = Sum_{k>=0} C(n*k+1,k)/(n*k+1)*x^(n*k+1),
%e A194560 G_n(x)^n = Sum_{k>=1} C(n*k,k)/(n*k-k+1)*x^(n*k);
%e A194560 the first few expansions of G_n(x)^n begin:
%e A194560 G_1(x) = x + x^2 + x^3 + x^4 + x^5 + ...
%e A194560 G_2(x)^2 = x^2 + 2*x^4 + 5*x^6 + 14*x^8 + ... + A000108(n)*x^(2*n) + ...
%e A194560 G_3(x)^3 = x^3 + 3*x^6 + 12*x^9 + 55*x^12 + ... + A001764(n)*x^(3*n) + ...
%e A194560 G_4(x)^4 = x^4 + 4*x^8 + 22*x^12 + 140*x^16 + ... + A002293(n)*x^(4*n) + ...
%e A194560 G_5(x)^5 = x^5 + 5*x^10 + 35*x^15 + 285*x^20 + ... + A002294(n)*x^(5*n) + ...
%t A194560 Table[Sum[Binomial[n,d]/(n-d+1),{d,Divisors[n]}],{n,20}] (* _Gus Wiseman_, Feb 15 2019 *)
%o A194560 (PARI) {a(n)=if(n<1,0,sumdiv(n,d,binomial(n,d)/(n-d+1)))}
%o A194560 (PARI) {a(n)=polcoeff(sum(m=1,n,serreverse(x/(1+x^m+x*O(x^n)))^m),n)}
%Y A194560 Row sums of A306437.
%Y A194560 Cf. A000108, A001263, A001764, A016098, A038041, A061095, A125181, A134264, A194558, A306418, A306438.
%K A194560 nonn
%O A194560 1,2
%A A194560 _Paul D. Hanna_, Aug 28 2011