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\\ 1) 1 / ( (1-x)^u * (1-x+x^s)^t )^n
\\ Cf. A372462
a(n, s, t, u) = sum(k=0, n\s, (-1)^k*binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( 1 / ( (1-x)^u * (1-x+x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ 2) 1 / ( (1-x)^u * (1-x-x^s)^t )^n
\\ Cf. A372233
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( 1 / ( (1-x)^u * (1-x-x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ 3) ( (1-x+x^s)^t / (1-x)^u )^n
\\ 実質的に1)と同じ
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n, k)*binomial((u-t+1)*n-(s-1)*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( (1-x+x^s)^t / (1-x)^u )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ 4) ( (1-x)^u / (1-x-x^s)^t )^n
\\ 実質的に2)と同じ
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( (1-x)^u / (1-x-x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ 5) ( (1+x)^u * (1+x+x^s)^t )^n
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( (1+x)^u * (1+x+x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
\\ 1/( (1-x)^u * (1-x^s)^t )^n
\\ Cf. A370274
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( 1/( (1-x)^u * (1-x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ ( 1/(1-x)^u * (1+x^s)^t )^n
\\ Cf. A370244
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n, k)*binomial((u+1)*n-s*k-1, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( 1/(1-x)^u * (1+x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ ( (1+x)^u / (1+x^s)^t )^n
\\ Cf. A370214
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial(u*n, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( (1+x)^u / (1-x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");
\\ ( (1+x)^u * (1+x^s)^t )^n
\\ Cf. A370194
a(n, s, t, u) = sum(k=0, n\s, binomial(t*n, k)*binomial(u*n, n-s*k));
b(n, s, t, u) = my(x='x+O('x^(n+2))); polcoef( ( (1+x)^u * (1+x^s)^t )^n, n);
for(s=1, 10, for(t=-10, 10, for(u=-20, 20, for(n=0, 20, if(a(n, s, t, u)!=b(n, s, t, u), print1([n, s, t, u], ", "))))));
print1("done");