Original entry on oeis.org
1, 0, 1, 1, 2, 3, 7, 12, 27, 55, 128, 285
Offset: 1
- Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
- Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
A283827
Irregular triangle read by rows: T(n,k) = number of trees on n nodes with load k, n>=1, 1 <= k <= floor(n/2).
Original entry on oeis.org
0, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 3, 7, 1, 3, 9, 10, 1, 4, 12, 30, 1, 4, 18, 38, 45, 1, 5, 21, 64, 144, 1, 5, 27, 91, 217, 210
Offset: 1
Triangle begins:
0,
1,
1,
1, 1,
1, 2,
1, 2, 3,
1, 3, 7,
1, 3, 9, 10,
1, 4, 12, 30,
1, 4, 18, 38, 45,
1, 5, 21, 64, 144,
1, 5, 27, 91, 217, 210,
...
- Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
A328057
Number of graphs with n nodes having fewer than n edges.
Original entry on oeis.org
1, 2, 3, 7, 14, 33, 81, 215, 601, 1808, 5721, 19133, 67218, 247377, 950679, 3806360, 15837196, 68336348, 305196782, 1408294018, 6703197359, 32861879994, 165699114887, 858237346563, 4560774579700, 24839216194151, 138505159164086, 789982051646096, 4604866422703625
Offset: 1
-
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^g, {i, 2, Length[v]}, {j, 1, i - 1}]*Product[c = v[[i]]; t[c]^Quotient[c - 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
a[n_] := a[n] = Module[{s = O[x]^n}, Do[s += permcount[p]*edges[p, 1 + x^# + O[x]^n &], {p, IntegerPartitions[n]}]; SeriesCoefficient[s/(1-x), {x, 0, n - 1}]/n!];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 08 2021, after Andrew Howroyd *)
-
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}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
a(n)={my(s=O(x^n)); forpart(p=n, s+=permcount(p)*edges(p, i->1 + x^i + O(x^n))); polcoef(s/(1-x), n-1)/n!} \\ Andrew Howroyd, Oct 22 2019
A005143
Number of sub-Eulerian graphs with n nodes.
Original entry on oeis.org
1, 1, 3, 10, 45, 274
Offset: 2
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
A005144
Number of sub-Hamiltonian graphs with n nodes.
Original entry on oeis.org
1, 1, 2, 10, 43, 346
Offset: 2
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
A120412
Number of unlabeled graphs with n equal to the number of vertices plus the number of edges.
Original entry on oeis.org
1, 1, 2, 2, 3, 5, 7, 10, 16, 25, 40, 66, 111, 191, 343, 627, 1182, 2301, 4609, 9511, 20229, 44252, 99564, 230171, 546118, 1328476, 3309876, 8436887, 21980376, 58473130, 158692559, 439012704, 1237049733, 3547984011, 10350963267, 30699209481, 92508993842
Offset: 1
Petr Vojtechovsky (petr(AT)math.du.edu), Jul 05 2006
a(3) = 2 because there is a graph with 3 vertices and no edges and a graph with 2 vertices and one edge.
-
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^g, {j, 1, i - 1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[c-1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := row[n] = Module[{s = 0}, Do[s += permcount[p]*edges[p, 1+x^#&], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x]&;
T[n_, k_] := If[k <= Length[row[n]], row[n][[k]], 0];
a[n_] := Sum[T[k, n-k+1], {k, 1, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 37}] (* Jean-François Alcover, Jan 09 2021, after Andrew Howroyd in A008406 *)
-
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}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
seq(n)={Vec(sum(k=1, n, x^k*G(k, x + O(x*x^(n-k)))))} \\ Andrew Howroyd, Nov 07 2019
A171534
Triangle read by rows: T(n,k)=binomial(2*n*k+k,2)/(2*n+1).
Original entry on oeis.org
1, 2, 9, 3, 13, 30, 4, 17, 39, 70, 5, 21, 48, 86, 135, 6, 25, 57, 102, 160, 231, 7, 29, 66, 118, 185, 267, 364, 8, 33, 75, 134, 210, 303, 413, 540, 9, 37, 84, 150, 235, 339, 462, 604, 765, 10, 41, 93, 166, 260, 375, 511, 668, 846, 104
Offset: 1
1;
2, 9;
3, 13, 30;
4, 17, 39, 70;
5, 21, 48, 86, 135;
6, 25, 57, 102, 160, 231;
7, 29, 66, 118, 185, 267, 364;
8, 33, 75, 134, 210, 303, 413, 540;
9, 37, 84, 150, 235, 339, 462, 604, 765;
10, 41, 93, 166, 260, 375, 511, 668, 846, 1045;
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 145.
-
Table[Table[Binomial[2*n*k + k, 2]/(2*n + 1), {k, 1, n}], {n, 1, 10}]
Flatten[%]
Introduced OEIS notational standards and keyword:tabl - The Assoc. Editors of the OEIS, Dec 15 2009
A182043
Table, by rows, of T(k,n) the number of simple graphs on v = prime(n) vertices and with e = prime(k) edges.
Original entry on oeis.org
1, 1, 2, 4, 6, 4, 2, 5, 21, 65, 148, 97, 10, 2, 2, 5, 26, 172, 10250, 75415, 2295898, 8640134, 53037356, 99187806, 70065437, 4609179, 192788, 28259, 467, 2, 2, 5, 26, 176, 14140, 154658, 17422984, 152339952, 6461056816, 359954668522, 899632282299, 4093273437761, 4093273437761
Offset: 2
T(3,4) = 4 because there are 4 simple graphs with prime(3) = 5 vertices and prime(4) = 7 edges.
The table begins:
+---+---+---+---+
|e=2|e=3|e=5|e=7|
+---+---+---+---+---+
|v=3| 1 | 1 | | |
+---+---+---+---+---+
|v=5| 2 | 4 | 6 | 4 |
+---+---+---+---+---+
-
read("transforms3") :
L := BFILETOLIST("b008406.txt") ;
A008406 := proc(n,k)
global L ;
local f,r ;
f := 1 ;
r := 1 ;
while r < n do
f := f+r*(r-1)/2+1 ;
r := r+1 ;
end do:
op(f+k,L) ;
end proc:
for n from 1 do
v := ithprime(n) ;
for k from 1 do
e := ithprime(k) ;
if e > v*(v-1)/2 then
break;
else
printf("%d,",A008406(v,e)) ;
end if;
end do:
end do: # R. J. Mathar, Oct 20 2013
A283822
Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) = number of graphs with n edges, no isolated nodes, and k nodes in largest connected component.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 7, 8, 6, 1, 6, 15, 16, 19, 11, 1, 7, 24, 34, 44, 44, 23, 1, 9, 38, 71, 97, 122, 112, 47, 1, 11, 61, 133, 211, 295, 371, 287, 106, 1, 13, 90, 249, 457, 659, 1015, 1131, 763, 235
Offset: 1
Triangle begins:
1,
1,1,
1,2,2,
1,3,4,3,
1,4,7,8,6,
1,6,15,16,19,11,
1,7,24,34,44,44,23,
1,9,38,71,97,122,112,47,
1,11,61,133,211,295,371,287,106,
1,13,90,249,457,659,1015,1131,763,235,
...
- Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
A283824
Number of unlabeled (and unrooted) trees on n nodes having a node that is both a center and a centroid.
Original entry on oeis.org
1, 1, 0, 1, 1, 2, 2, 6, 7, 20, 27, 83, 126
Offset: 0
- Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
- Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Comments