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 A303875 Number of noncrossing partitions of an n-set up to rotation and reflection with all blocks having a prime number of elements. 5

%I

%S 1,0,1,1,1,2,3,5,7,14,26,49,107,215,502,1112,2619,6220,14807,36396,

%T 88397,219920,545196,1364669,3434436,8658463,21989434,55893852,

%U 142823174,365766327,939575265,2420885031,6250344302,16183450744,41981605437,109155492638

%N Number of noncrossing partitions of an n-set up to rotation and reflection with all blocks having a prime number of elements.

%C The number of such noncrossing partitions counted distinctly is given by A210737.

%H Andrew Howroyd, <a href="/A303875/b303875.txt">Table of n, a(n) for n = 0..500</a>

%o (PARI) \\ number of partitions with restricted block sizes

%o NCPartitionsModDihedral(v)={ my(n=#v);

%o my(p=serreverse( x/(1 + sum(k=1, #v, x^k*v[k])) + O(x^2*x^n) )/x);

%o my(vars=variables(p));

%o my(varpow(r,d)=substvec(r + O(x^(n\d+1)), vars, apply(t->t^d, vars)));

%o my(q=x*deriv(p)/p, h=varpow(p,2));

%o my(R=sum(i=0, (#v-1)\2, v[2*i+1]*x*(x^2*h)^i), Q=sum(i=1, #v\2, v[2*i]*(x^2*h)^i), T=sum(k=1, #v, my(t=v[k]); if(t, x^k*t*sumdiv(k, d, eulerphi(d) * varpow(p,d)^(k/d))/k)));

%o (T + 2 + intformal(sum(d=1, n, eulerphi(d)*varpow(q,d))/x) - p + (1 + Q + (1+R)^2*h/(1-Q))/2)/2 + O(x*x^n)

%o }

%o Vec(NCPartitionsModDihedral(vector(40,k,isprime(k))))

%Y Cf. A111275, A185100 (1 or 2), A210737, A303874, A303931.

%K nonn

%O 0,6

%A _Andrew Howroyd_, May 01 2018

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Last modified February 22 08:35 EST 2020. Contains 332133 sequences. (Running on oeis4.)