Revision History for A283877
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#29 by Michael De Vlieger at Tue Jan 16 17:42:29 EST 2024
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#28 by Andrew Howroyd at Tue Jan 16 17:42:11 EST 2024
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#27 by Andrew Howroyd at Tue Jan 16 17:35:56 EST 2024
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| PROG
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(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t), x, x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g, x, x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024
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Discussion
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| Tue Jan 16
| 17:42
| Andrew Howroyd: Other sequences that referred to this function have already been updated.
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#26 by Andrew Howroyd at Tue Jan 16 17:34:54 EST 2024
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(PARI) \\ SetTypes function referenced by other sequences.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
V(n, w)={sumdiv(gcd(n, w), d, moebius(d)*binomial(n/d, w/d))/n}
S(n)={my(v=vector(n)); for(w=0, n, fordiv(gcd(n, w), d, v[n/d] += x^w*V(n/d, w/d))); v}
SetTypes(ptyp, fx)={
my(lim=sum(i=1, #ptyp, ptyp[i]), u=vector(lim, i, O(x*x^(lim\i)))); u[1] += 1;
for(i=1, #ptyp, my(s=S(ptyp[i]), v=vector(#u)); for(j=1, #u, for(k=1, #s, my(g=lcm(j, k)); if(g<=#v, v[g]+=u[j]*s[k]*j*k/g))); u=v);
u[1]-=1; Vec(sum(i=1, #u, subst(fx(u[i]), x, x^i)) + O(x*x^lim), -lim); }
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*WeighT(SetTypes(p, q->q))[n]); s/n!} \\ Andrew Howroyd, Sep 01 2019
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| STATUS
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approved
editing
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#25 by OEIS Server at Mon Sep 02 14:45:52 EDT 2019
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| LINKS
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Andrew Howroyd, <a href="/A283877/b283877_1.txt">Table of n, a(n) for n = 0..50</a>
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#24 by N. J. A. Sloane at Mon Sep 02 14:45:52 EDT 2019
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Discussion
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| Mon Sep 02
| 14:45
| OEIS Server: Installed new b-file as b283877.txt. Old b-file is now b283877_1.txt.
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#23 by Michel Marcus at Mon Sep 02 11:52:03 EDT 2019
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#22 by Andrew Howroyd at Mon Sep 02 11:31:36 EDT 2019
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#21 by Andrew Howroyd at Sun Sep 01 23:06:41 EDT 2019
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(PARI) \\ codeSetTypes function sharedreferenced withby other sequences.
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#20 by Andrew Howroyd at Sun Sep 01 23:05:15 EDT 2019
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(PARI) \\ code shared with other sequences.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
V(n, w)={sumdiv(gcd(n, w), d, moebius(d)*binomial(n/d, w/d))/n}
S(n)={my(v=vector(n)); for(w=0, n, fordiv(gcd(n, w), d, v[n/d] += x^w*V(n/d, w/d))); v}
SetTypes(ptyp, fx)={
my(lim=sum(i=1, #ptyp, ptyp[i]), u=vector(lim, i, O(x*x^(lim\i)))); u[1] += 1;
for(i=1, #ptyp, my(s=S(ptyp[i]), v=vector(#u)); for(j=1, #u, for(k=1, #s, my(g=lcm(j, k)); if(g<=#v, v[g]+=u[j]*s[k]*j*k/g))); u=v);
u[1]-=1; Vec(sum(i=1, #u, subst(fx(u[i]), x, x^i)) + O(x*x^lim), -lim); }
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*WeighT(SetTypes(p, q->q))[n]); s/n!} \\ Andrew Howroyd, Sep 01 2019
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