OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..40
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Edge Cut.
PROG
(PARI)
B(m, n) = {16^binomial(m, 2) * 2^binomial(n, 2) * 4^(m*n)}
S(m, n)={ my(M=matrix(m+1, n+1, i, j, B(i-1, j-1)), N=matrix(m+1, n+1));
for(n=0, n, N[1, 1+n] = M[1, 1+n] - sum(j=1, n-1, binomial(n-1, j-1)*N[1, 1+j]*M[1, 1+n-j]));
for(m=1, m, for(n=0, n, N[1+m, 1+n] = M[1+m, 1+n] - sum(i=1, m, sum(j=0, n, binomial(m-1, i-1)*binomial(n, j)*N[1+i, 1+j]*M[1+m-i, 1+n-j]))));
N
}
seq(n)=my(M=S(n, 2*n)); vector(n, n, if(n==1, 0, 16^binomial(n, 2) - sum(k=0, n, (-1)^k*binomial(n, k)*M[1+k, 1+2*(n-k)]))) \\ Andrew Howroyd, Jun 12 2025
(PARI)
D(n, i)={16^binomial(i, 2)*sum(j=0, 2*(n-i), 2^binomial(j, 2)*4^(i*j)*x^j/j!, O(x^(2*(n-i)+1)))}
E(n)={my(u=vector(n+1, i, D(n, i-1)), v=vector(n+1)); v[1]=1+log(u[1]); for(m=1, n, v[1+m] = (u[1+m] - sum(i=1, m-1, binomial(m-1, i-1)*v[1+i]*u[1+m-i]))/u[1] ); v}
seq(n)={my(u=E(n)); vector(n, n, if(n==1, 0, 16^binomial(n, 2) - sum(k=0, n, (-1)^k*binomial(n, k)*(2*n-2*k)!*polcoef(u[1+k], 2*(n-k)) )))} \\ Andrew Howroyd, Jun 12 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Nov 03 2024
EXTENSIONS
a(5)-a(7) from Andrew Howroyd, May 27 2025
a(8) onwards from Andrew Howroyd, Jun 12 2025
STATUS
approved