A136512 Produced by same formula that gives A093934 (signed tournaments), but with LCM instead of GCD in the exponent.
1, 2, 4, 12, 64, 616, 10304, 293744, 14381056, 1242433312, 196990542848, 59624929814720, 35242762808786944, 40573409794074305152, 89317952471536946659328, 368970766373159503907450624, 2827862662172992194150488080384, 40061570271801436240253461050024448, 1050869620561002649814192493096912289792
Offset: 0
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..50
Crossrefs
Cf. A093934.
Programs
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PARI
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m} edges(v) = {sum(i=2, #v, sum(j=1, i-1, lcm(v[i], v[j]))) + sum(i=1, #v, v[i]\2)} oddp(v) = {for(i=1, #v, if(bitand(v[i], 1)==0, return(0))); 1} a(n) = {my(s=0); forpart(p=n, if(oddp(p), s+=permcount(p)*2^(#p+edges(p)))); s/n!} \\ Andrew Howroyd, Feb 29 2020
Formula
a(n) = Sum_{j} (1/(Product (k^(j_k) (j_k)!))) * 2^{t_j},
where j runs through all partitions of n into odd parts, say with j_1 parts of size 1, j_3 parts of size 3, etc.,
and t_j = (1/2)*[ Sum_{r=1..n, s=1..n} j_r j_s lcm(r,s) + Sum_{r} j_r ].