A098233
Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 4, 7, 3, 1, 1, 13, 46, 47, 25, 6, 1, 1, 40, 295, 587, 516, 235, 65, 10, 1, 1, 121, 1846, 6715, 9690, 7053, 3006, 800, 140, 15, 1, 1, 364, 11347, 73003, 170051, 189458, 119211, 46795, 12201, 2170, 266, 21, 1, 1, 1093, 68986, 768747
Offset: 0
The first few polynomials are:
1,
x^2,
x^2+x^3+x^4,
x^2+4x^3+7x^4+3x^5+x^6,
x^2+13x^3+46x^4+47x^5+25x^6+6x^7+x^8,
x^2+40x^3+295x^4+587x^5+516x^6+235x^7+65x^8+10x^9+x^10,
...
Triangle starts:
1;
1;
1, 1, 1;
1, 4, 7, 3, 1;
1, 13, 46, 47, 25, 6, 1;
1, 40, 295, 587, 516, 235, 65, 10, 1;
...
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
- Steve Butler, Fan Chung, Jay Cummings, and R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
- L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
A020558
Number of ordered multigraphs on n labeled edges (without loops).
Original entry on oeis.org
1, 1, 4, 27, 274, 3874, 71995, 1682448, 47840813, 1615315141, 63566760077, 2873099980637, 147384910116793, 8496500896980637, 545845612016485842, 38797966029876716897, 3032005571734589578076
Offset: 0
- G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
- 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]
A175707
Number of ways to put n copies of 1,2,3,4 into sets.
Original entry on oeis.org
1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756, 351989816, 575711716, 921889652, 1447822620, 2233501928, 3389114724, 5064582169, 7461570579, 10848490675, 15579077786, 22115241763, 31054971635, 43166197978, 59427633555, 81077755892, 109673237289, 147158299390, 195946638641
Offset: 0
For n=1, the solution is the fourth term of Bell numbers A000110.
For n=2, one way to partition 2 copies of 1, 2 copies of 2, 2 copies of 3 and 2 copies of 4 is {1}{2}{34}{12}{34}. On the other hand {112}{34}{23}{4} is not allowed since the same numbers are in the same set {112}.
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls? (2009).
- Doron Zeilberger, In How many ways can you reassemble several russian dolls?, arXiv:0909.3453 [math.CO], 2009.
- Doron Zeilberger, BABUSHKAS; Local copy
- Index entries for linear recurrences with constant coefficients, signature (7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1).
-
a:= n-> (5382*n^11 +236808*n^10 +4643760*n^9 +53507520*n^8 +402098796*n^7 +2067612624*n^6 +7421736960*n^5 +18616942080*n^4 +32101468047*n^3 +36555545268*n^2 +25131098880*n +8024016000 +7016625*(-1)^n*n^3 +84199500*(-1)^n*n^2 +359251200*(-1)^n*n +538876800*(-1)^n) /(2^11*3^7*5^2*7*11) +5/3^6*(-1)^n * (sin(n*Pi/3)/sqrt(3)+ cos(n*Pi/3));
seq(a(n), n=0..40);
seq(SeqBrnDJ(n,4)[5], n=1..6); # using the Maple package BABUSHKAS (see links)
-
LinearRecurrence[{7, -17, 8, 36, -60, 4, 56, -22, -22, -22, 56, 4, -60, 36, 8, -17, 7, -1}, {1, 15, 139, 862, 4079, 15791, 52450, 154279, 411180, 1009741, 2314278, 5000125, 10264997, 20152950, 38037517, 69323949, 122448455, 210271756}, 36] (* Jean-François Alcover, Nov 13 2018 *)
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