A215858
Number of simple labeled graphs on n nodes with exactly 8 connected components that are trees or cycles.
Original entry on oeis.org
1, 36, 1110, 31680, 904299, 26603148, 821278744, 26864874465, 935625630797, 34750489933016, 1375999952017938, 57998361908305494, 2596646585329104847, 123180358220543885268, 6175880603945440333627, 326438846760992348696038, 18147404450341079958539275
Offset: 8
a(9) = 36: each graph has one 2-node tree and 7 1-node trees and C(9,2) = 36.
-
T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
`if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
end:
a:= n-> T(n, 8):
seq(a(n), n=8..25);
A215859
Number of simple labeled graphs on n nodes with exactly 9 connected components that are trees or cycles.
Original entry on oeis.org
1, 45, 1650, 54945, 1795794, 59546487, 2043490735, 73415619420, 2779264615127, 111226656560877, 4710924208619304, 211105699482022215, 9997623229700175712, 499562336689773070263, 26288415481415803589236, 1454007169289989503463230, 84361156450441837460650255
Offset: 9
a(10) = 45: each graph has one 2-node tree and 8 1-node trees and C(10,2) = 45.
-
T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
`if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
end:
a:= n-> T(n, 9):
seq(a(n), n=9..25);
A215860
Number of simple labeled graphs on n nodes with exactly 10 connected components that are trees or cycles.
Original entry on oeis.org
1, 55, 2365, 90805, 3367364, 124984860, 4743643190, 186488038880, 7653850266777, 329429479792985, 14903545528332565, 709243144460040645, 35495878932860944422, 1866637759375098988740, 103014318586612720480259, 5957391569989223921495400
Offset: 10
a(11) = 55: each graph has one 2-node tree and 9 1-node trees and C(11,2) = 55.
-
T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(binomial(n-1, i)*T(n-1-i, k-1)*
`if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
end:
a:= n-> T(n, 10):
seq(a(n), n=10..30);
A215863
Number of simple labeled graphs on n+3 nodes with exactly n connected components that are trees or cycles.
Original entry on oeis.org
0, 19, 135, 540, 1610, 3990, 8694, 17220, 31680, 54945, 90805, 144144, 221130, 329420, 478380, 679320, 945744, 1293615, 1741635, 2311540, 3028410, 3920994, 5022050, 6368700, 8002800, 9971325, 12326769, 15127560, 18438490, 22331160, 26884440, 32184944, 38327520
Offset: 0
a(1) = 19:
.1-2. .1-2. .1 2. .1-2. .1-2. .1 2. .1 2. .1 2. .1-2. .1-2.
.| |. . X . .|X|. .|\ . . /|. . \|. .|/ . .| |. .| . .| |.
.4-3. .4-3. .4.3. .4.3. .4.3. .4-3. .4-3. .4-3. .4-3. .4.3.
.
.1-2. .1 2. .1-2. .1-2. .1-2. .1 2. .1 2. .1 2. .1 2.
. |. . X . . / . . \ . . X . .|/|. . X|. .|X . .|\|.
.4-3. .4-3. .4-3. .4-3. .4.3. .4.3. .4.3. .4.3. .4.3.
A215864
Number of simple labeled graphs on n+4 nodes with exactly n connected components that are trees or cycles.
Original entry on oeis.org
0, 137, 1267, 6412, 23597, 70707, 183099, 424809, 904299, 1795794, 3367364, 6017011, 10318126, 17075786, 27395466, 42765846, 65157498, 97139343, 142014873, 203980238, 288306403, 401547685, 551779085, 748864935, 1004761485, 1333856160, 1753346322, 2283660477
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
-
a:= n-> binomial(n+4,5)*(3886+(2285+(390+15*n)*n)*n)/48:
seq(a(n), n=0..40);
-
CoefficientList[Series[(7x^3+59x^2-34x-137)x/(x-1)^9,{x,0,30}],x] (* Harvey P. Dale, Jul 18 2019 *)
A215865
Number of simple labeled graphs on n+5 nodes with exactly n connected components that are trees or cycles.
Original entry on oeis.org
0, 1356, 15029, 90734, 394506, 1381695, 4138827, 11002068, 26603148, 59546487, 124984860, 248436188, 471271892, 858408642, 1508851218, 2569865520, 4255708464, 6872006526, 10847057991, 16771536474, 25448295950, 37954221305, 55716334245, 80604653220
Offset: 0
A309313
Number of simple labeled graphs on 2n nodes with exactly n connected components that are trees or cycles.
Original entry on oeis.org
1, 1, 19, 540, 23597, 1381695, 101682724, 9016296289, 935625630797, 111226656560877, 14903545528332565, 2222230881719482634, 364942065096639623872, 65448490334085989020670, 12726830901257817750060165, 2667188536603107740647377075, 599286881811684624273478547325
Offset: 0
-
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, add(binomial(n-1, i)*b(n-1-i, k-1)*
`if`(i<2, 1, i!/2 +(i+1)^(i-1)), i=0..n-k)))
end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);
-
b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0,
If[n == 0, 1, Sum[Binomial[n - 1, i]*b[n - 1 - i, k - 1]*
If[i<2, 1, i!/2 + (i+1)^(i-1)], {i, 0, n-k}]]];
a[n_] := b[2n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
Comments