A276549
Number of primitive (aperiodic) reversible string structures with n beads using an infinite alphabet.
Original entry on oeis.org
1, 1, 3, 9, 31, 112, 467, 2141, 10739, 58454, 340389, 2110093, 13830234, 95475087, 691543059, 5240282987, 41432986587, 341040306207, 2916376237349, 25862097428262, 237434959190586, 2253358056942644, 22076003468637449, 222979436688500085, 2319295172178428701
Offset: 1
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b[n_] := SeriesCoefficient[Exp[(Exp[2*x] - 3)/2 + Exp[x]], {x, 0, n}]*n!;
c[n_] := If[n == 0, 1, (BellB[n - 1] + If[Mod[n, 2] == 1, b[(n - 1)/2], Sum[Binomial[n/2 - 1, k]*b[k], {k, 0, n/2 - 1}]])/2];
a[n_] := DivisorSum[n, MoebiusMu[n/#] c[# + 1]&];
Array[a, 25] (* Jean-François Alcover, Jun 16 2017, using Alois P. Heinz's code for A103293 *)
A327612
Number of length n reversible string structures that are not palindromic using any number of colors.
Original entry on oeis.org
0, 1, 2, 9, 27, 112, 453, 2137, 10691, 58435, 340187, 2110016, 13829358, 95474679, 691538954, 5240280999, 41432965441, 341040295916, 2916376121375, 25862097370783, 237434958512487, 2253358056604465, 22076003464423853, 222979436686398848, 2319295172150784296
Offset: 1
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\\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
seq(n)={my(A=Ach(n)); vector(n, i, sum(k=1, n, (A[i,k] + stirling(i, k, 2))/2 - stirling((i+1)\2, k, 2)))}
A376620
Number of equational laws for magmas involving n operations, up to relabeling and symmetry.
Original entry on oeis.org
2, 5, 41, 364, 4294, 57882, 888440, 15120105, 281942218, 5698630860, 123850400282, 2875187314622, 70909556575040, 1849319825544900, 50801676938400207, 1464954360561398340, 44213852151914127210, 1392971702129279452950, 45705100441643456206404, 1558551328538087579977710
Offset: 0
For n=0 the distinct laws are x=x and x=y.
For n=1 the distinct laws are x=x*x, x=x*y, x=y*x, x=y*y, and x=y*z. (x*y=z, for instance, is a relabeling of x=y*z after applying symmetry.)
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\\ All functions that are needed
a110(n) = sum(k=0, n, stirling(n,k,2)); \\ Bell
a108(n) = binomial(2*n,n)/(n+1); \\ Catalan
a289679(n) = a108(n-1)*a110(n);
Ach(n,k)= my(s=n<2 && n>=0 && n==k); if(n<=1, s, k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2) + s);
a103293(n) = if(n<3, 1, sum(k=0, n-1, stirling(n-1,k,2) + Ach(n-1,k))/2);
a376620(n) = if(n%2==0,(a289679(n+2) + a108(n/2) * (2*a103293(n+3) - a110(n+2)))/2, a289679(n+2)/2); \\ Hugo Pfoertner, Sep 30 2024
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from functools import lru_cache
from sympy.functions.combinatorial.numbers import stirling, bell, catalan
def A376620(n):
if n&1:
return catalan(n+1)*bell(n+2)>>1
else:
@lru_cache(maxsize=None)
def ach(n,k): return (n==k) if n<2 else k*ach(n-2,k)+ach(n-2,k-1)+ach(n-2,k-2)
return catalan(n+1)*bell(n+2)+catalan(n>>1)*((sum(stirling(n+2,k,kind=2)+ach(n+2,k)>>1 for k in range(n+3))<<1)-bell(n+2))>>1 # Chai Wah Wu, Oct 15 2024
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