Search: a098622 -id:a098622
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A098620
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Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges.
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+10
13
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1, 1, 4, 26, 257, 3586, 66207, 1540693, 43659615, 1469677309, 57681784820, 2601121752854, 133170904684965, 7664254746784243, 491679121677763607, 34905596059311761907, 2725010800987216480527, 232643959843709167832482, 21613761720729431904201734
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0,3
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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FORMULA
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PROG
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(PARI) \\ 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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A098623
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Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.
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+10
8
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1, 1, 8, 109, 2229, 62684, 2289151, 104344153, 5767234550, 378073098155, 28888082263581, 2536660090249102, 253007765488793325, 28383529110762969901, 3551558435250676339536, 492092920443604792460905, 75025155137863150912784409, 12516480979952118669729618300
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OFFSET
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0,3
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ 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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A098626
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Consider the family of directed multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
3
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1, 0, 2, 4, 57, 348, 5235, 57930, 1037540, 16842496, 363889755, 7792175070, 201054289293, 5345844537876, 162234861271288, 5156725529935952, 181284205622239755, 6713109719185427600, 269652617328843102055, 11418447984579685481310, 517839485352765454438270
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OFFSET
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0,3
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is A000166 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(-x + O(x*x^n))/(1-x)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A098630
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Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled loops and arcs
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+10
3
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1, 4, 60, 1624, 66240, 3711200, 269670208, 24435113216, 2682916389632, 349223324753408, 52965538033020928, 9229753832340117504, 1826647528631522463744, 406579171521484851396608, 100934277604965329345822720, 27746271707522968205726416896
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graph;
refs;
listen;
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text;
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) a(n) = {2^n*sum(k=0, 2*n, stirling(2*n, k, 2))} \\ Andrew Howroyd, Jan 12 2021
(PARI) \\ R(n) is A000079 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(2*x + O(x*x^n))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A098638
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Consider the family of directed multigraphs enriched by the species of odd sets. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
3
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1, 2, 13, 164, 3127, 82600, 2845775, 122820136, 6446913953, 402413160952, 29343933156485, 2464029760993520, 235446319553848087, 25346231173047308256, 3047931031445529965527, 406412844141860523543392, 59704680455100785101683457, 9608818815170839730520275488
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graph;
refs;
listen;
history;
text;
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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E.g.f.: exp(-1)*Sum_{n>=0}(1+sinh(x))^(n^2)/n!. - Vladeta Jovovic, Mar 04 2008
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PROG
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(PARI) \\ EnrichedGdlSeq defined in A098622.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Missing a(10) inserted and terms a(13) and beyond from Andrew Howroyd, Jan 12 2021
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STATUS
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approved
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A099694
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Consider the family of directed multigraphs enriched by the species of directed sets. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
3
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1, 2, 17, 244, 5283, 156092, 5954547, 282221828, 16159327961, 1094056231572, 86116276633357, 7773114989571400, 795480206815177651, 91417037615848058160, 11701283925663217478843, 1656436690705751478232180, 257730676653629520748175377, 43837005194184348815823808500
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is e.g.f. of 1, 1, 2, 2, 2, ...; EnrichedGdlSeq defined in A098622.
R(n)={2*exp(x + O(x*x^n)) - x - 1}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A099698
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Consider the family of directed multigraphs enriched by the species of involutions. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
3
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1, 2, 17, 248, 5403, 160420, 6142567, 291996934, 16759322733, 1136940595762, 89641455771637, 8102778995663368, 830222723124364047, 95509354134959796556, 12236166882713532940611, 1733521075683722202738222, 269910543278748394820341769, 45936441912756036235229989058
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listen;
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OFFSET
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0,2
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is A000085 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(x+x^2/2 + O(x*x^n))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Dead sequence restored, corrected and extended by Andrew Howroyd, Jan 12 2021
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STATUS
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approved
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A099702
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Consider the family of directed multigraphs enriched by the species of simple graphs. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
2
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1, 2, 17, 256, 5719, 173446, 6768075, 328288840, 19468007553, 1458080017522, 183476204746761, 87209577493989776, 154656821805639801687, 617619828457724835488214, 5008102331929281541386123923, 81618549234469098721106601012472, 2666950050438611111026601803629686849
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is A006125 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={sum(k=0, n, 2^binomial(k, 2)*x^k/k!) + O(x*x^n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A099706
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Consider the family of directed multigraphs enriched by the species of directed graphs. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
2
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1, 4, 84, 3568, 305712, 87782720, 144600947392, 1139235294403328, 37012349010095737088, 4840037457225169875031040, 2535930555678883610642223895552, 5317274645187046706095607711946092544, 44602319906972740832371696997145322907873280
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is A002416 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={sum(k=0, n, 2^(k^2)*x^k/k!) + O(x*x^n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A099710
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Consider the family of directed multigraphs enriched by the species of endofunctions. Sequence gives number of those multigraphs with n labeled loops and arcs.
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+10
2
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1, 2, 21, 372, 9503, 323528, 13976119, 740471952, 46918236113, 3486842393336, 299252510858253, 29285226572514608, 3233515108614711055, 399237909648934968160, 54699907257463871118015, 8261287679602024304387776, 1367355850924129919137226337, 246745297507913180076213875232
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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REFERENCES
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G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
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LINKS
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FORMULA
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PROG
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(PARI) \\ R(n) is A000312 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={1/(1 + lambertw(-x + O(x*x^n)))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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