A098620
Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges.
Original entry on oeis.org
1, 1, 4, 26, 257, 3586, 66207, 1540693, 43659615, 1469677309, 57681784820, 2601121752854, 133170904684965, 7664254746784243, 491679121677763607, 34905596059311761907, 2725010800987216480527, 232643959843709167832482, 21613761720729431904201734
Offset: 0
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
-
\\ here R(n) is A000110 as e.g.f.
egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i, k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
EnrichedGnSeq(R)={my(n=serprec(R, x)-1, B=exp(x/2 + O(x*x^n))*subst(egf1(n), x, log(1+x + O(x*x^n))/2)); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
A098622
Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled loops and arcs.
Original entry on oeis.org
1, 2, 17, 250, 5465, 162677, 6241059, 297132409, 17075153860, 1159545515804, 91501467848088, 8276847825732141, 848577193578286942, 97672164219292005480, 12518933902769241287267, 1774279753092963892540493, 276351502436571180980604240, 47046745370508674770872396843
Offset: 0
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
-
\\ here R(n) is A000110 as e.g.f.
egfA014507(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, stirling(i,k,1)*polcoef(bell, 2*k))*x^i/i!) + O(x*x^n)}
EnrichedGdlSeq(R)={my(n=serprec(R, x)-1); Vec(serlaplace(subst(egfA014507(n), x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
A098623
Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.
Original entry on oeis.org
1, 1, 8, 109, 2229, 62684, 2289151, 104344153, 5767234550, 378073098155, 28888082263581, 2536660090249102, 253007765488793325, 28383529110762969901, 3551558435250676339536, 492092920443604792460905, 75025155137863150912784409, 12516480979952118669729618300
Offset: 0
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
-
\\ here R(n) is A000110 as e.g.f.
egfA020556(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i, k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
EnrichedGdSeq(R)={my(n=serprec(R, x)-1, B=subst(egfA020556(n), x, log(1+x + O(x*x^n)))); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
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