A320663 - OEIS

A320663

Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

1, 1, 4, 7, 21, 40, 106, 216, 534, 1139, 2715, 5962, 14012, 31420, 73484, 167617, 392714, 908600, 2140429, 5015655, 11905145, 28228533, 67590229, 162067916, 391695348, 949359190, 2316618809, 5673557284, 13979155798, 34583650498, 86034613145, 214948212879

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OFFSET

0,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50

EXAMPLE

Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:

{{1}} {{1,1}} {{1},{1,1}} {{1,1},{1,1}}

{{1,2}} {{1},{2,2}} {{1,1},{2,2}}

{{1},{1}} {{1},{2,3}} {{1,2},{1,2}}

{{1},{2}} {{2},{1,2}} {{1,2},{2,2}}

{{1},{1},{1}} {{1,2},{3,3}}

{{1},{2},{2}} {{1,2},{3,4}}

{{1},{2},{3}} {{1,3},{2,3}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{1},{2,3}}

{{1},{2},{1,2}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{2},{3,4}}

{{1},{3},{2,3}}

{{2},{2},{1,2}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{2},{2}}

{{1},{2},{3},{3}}

{{1},{2},{3},{4}}

PROG

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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}

gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i], v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((1+r)\2)*x^(2*r))}

a(n)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!} \\ Andrew Howroyd, Oct 26 2018

CROSSREFS

Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.

Sequence in context: A359603 A255512 A039959 * A186335 A010363 A119561

Adjacent sequences: A320660 A320661 A320662 * A320664 A320665 A320666

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 18 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Oct 26 2018

STATUS

approved