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Search: a098622 -id:a098622
Displaying 1-10 of 12 results found. page 1 2
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A098620 Consider the family of multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled edges. +10
13
1, 1, 4, 26, 257, 3586, 66207, 1540693, 43659615, 1469677309, 57681784820, 2601121752854, 133170904684965, 7664254746784243, 491679121677763607, 34905596059311761907, 2725010800987216480527, 232643959843709167832482, 21613761720729431904201734 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
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]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A000110. - Andrew Howroyd, Jan 12 2021
PROG
(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)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000110, A014500, A098621, A098622, A098623.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs. +10
8
1, 1, 8, 109, 2229, 62684, 2289151, 104344153, 5767234550, 378073098155, 28888082263581, 2536660090249102, 253007765488793325, 28383529110762969901, 3551558435250676339536, 492092920443604792460905, 75025155137863150912784409, 12516480979952118669729618300 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
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]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014505 and 1 + R(x) is the e.g.f. of A000110. - Andrew Howroyd, Jan 12 2021
PROG
(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)}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000110, A098620, A098621, A098622.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A098626 Consider the family of directed multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled loops and arcs. +10
3
1, 0, 2, 4, 57, 348, 5235, 57930, 1037540, 16842496, 363889755, 7792175070, 201054289293, 5345844537876, 162234861271288, 5156725529935952, 181284205622239755, 6713109719185427600, 269652617328843102055, 11418447984579685481310, 517839485352765454438270 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000166. - Andrew Howroyd, Jan 12 2021
PROG
(PARI) \\ R(n) is A000166 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(-x + O(x*x^n))/(1-x)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000166, A014507, A098622, A098624, A098625, A098627.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(14) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A098630 Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled loops and arcs +10
3
1, 4, 60, 1624, 66240, 3711200, 269670208, 24435113216, 2682916389632, 349223324753408, 52965538033020928, 9229753832340117504, 1826647528631522463744, 406579171521484851396608, 100934277604965329345822720, 27746271707522968205726416896 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
a(n) = 2^n*Bell(2*n). - Vladeta Jovovic, Aug 22 2006
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000079. - Andrew Howroyd, Jan 12 2021
PROG
(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))}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000079, A014507, A098622, A098628, A098629, A098631.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A098638 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. +10
3
1, 2, 13, 164, 3127, 82600, 2845775, 122820136, 6446913953, 402413160952, 29343933156485, 2464029760993520, 235446319553848087, 25346231173047308256, 3047931031445529965527, 406412844141860523543392, 59704680455100785101683457, 9608818815170839730520275488 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: exp(-1)*Sum_{n>=0}(1+sinh(x))^(n^2)/n!. - Vladeta Jovovic, Mar 04 2008
E.g.f.: B(sinh(x)) where B(x) is the e.g.f. of A014507. - Andrew Howroyd, Jan 12 2021
PROG
(PARI) \\ EnrichedGdlSeq defined in A098622.
EnrichedGdlSeq(sinh(x + O(x*x^20))) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A014507, A098622, A098636, A098637, A098639.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Missing a(10) inserted and terms a(13) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A099694 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. +10
3
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)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(2*exp(x) - x - 2) where B(x) is the e.g.f. of A014507. - Andrew Howroyd, Jan 12 2021
PROG
(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}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A014507, A098622, A099692, A099693, A099695.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A099698 Consider the family of directed multigraphs enriched by the species of involutions. Sequence gives number of those multigraphs with n labeled loops and arcs. +10
3
1, 2, 17, 248, 5403, 160420, 6142567, 291996934, 16759322733, 1136940595762, 89641455771637, 8102778995663368, 830222723124364047, 95509354134959796556, 12236166882713532940611, 1733521075683722202738222, 269910543278748394820341769, 45936441912756036235229989058 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000085. - Andrew Howroyd, Jan 12 2021
PROG
(PARI) \\ R(n) is A000085 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={exp(x+x^2/2 + O(x*x^n))}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000085, A014507, A098622, A099696, A099697, A099699.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 25 2017
EXTENSIONS
Dead sequence restored, corrected and extended by Andrew Howroyd, Jan 12 2021
STATUS
approved
A099702 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. +10
2
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)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A006125. - Andrew Howroyd, Jan 12 2021
PROG
(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)}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A006125, A014507, A098622, A099700, A099701, A099703.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A099706 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. +10
2
1, 4, 84, 3568, 305712, 87782720, 144600947392, 1139235294403328, 37012349010095737088, 4840037457225169875031040, 2535930555678883610642223895552, 5317274645187046706095607711946092544, 44602319906972740832371696997145322907873280 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A002416. - Andrew Howroyd, Jan 12 2021
PROG
(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)}
EnrichedGdlSeq(R(15)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A002416, A014507, A098622, A099704, A099705, A099707.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
A099710 Consider the family of directed multigraphs enriched by the species of endofunctions. Sequence gives number of those multigraphs with n labeled loops and arcs. +10
2
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)
OFFSET
0,2
REFERENCES
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
FORMULA
E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014507 and 1 + R(x) is the e.g.f. of A000312. - Andrew Howroyd, Jan 12 2021
PROG
(PARI) \\ R(n) is A000312 as e.g.f.; EnrichedGdlSeq defined in A098622.
R(n)={1/(1 + lambertw(-x + O(x*x^n)))}
EnrichedGdlSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021
CROSSREFS
Cf. A000312, A014507, A098622, A099708, A099709, A099711.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 26 2004
EXTENSIONS
Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021
STATUS
approved
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Last modified August 29 22:07 EDT 2024. Contains 375518 sequences. (Running on oeis4.)