A240682
Number of forests with n labeled nodes and 5 trees.
Original entry on oeis.org
1, 15, 210, 3220, 55755, 1092105, 24048255, 590412240, 16027796070, 477411574640, 15495339234375, 544652100894720, 20619226977792170, 836670560604157440, 36232055577668433690, 1668081561600000000000, 81363801140161673297535, 4191692026268767965880320
Offset: 5
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 5):
seq(a(n), n=5..30);
-
Table[n^(n-10) * (n-4)*(n-3)*(n-2)*(n-1)*(n^4 + 30*n^3 + 451*n^2 + 3846*n + 15120)/384,{n,5,20}] (* Vaclav Kotesovec, Sep 06 2014 *)
A240683
Number of forests with n labeled nodes and 6 trees.
Original entry on oeis.org
1, 21, 378, 7056, 143325, 3207897, 79170399, 2146836978, 63641666088, 2051450651250, 71530799628288, 2684845732979592, 107992630908804096, 4636019437800293718, 211623646464000000000, 10237455825414473977524, 523244238837133507448832, 28177157277452320985386539
Offset: 6
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 6):
seq(a(n), n=6..30);
-
Table[n^(n-12) * (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^5 + 40*n^4 + 835*n^3 + 10960*n^2 + 87636*n + 332640)/3840,{n,6,25}] (* Vaclav Kotesovec, Sep 06 2014 *)
A240684
Number of forests with n labeled nodes and 7 trees.
Original entry on oeis.org
1, 28, 630, 14070, 331485, 8411634, 231354123, 6899167275, 222569372025, 7741879425280, 289297137120992, 11570476164077376, 493535471267193810, 22376155441920000000, 1074961750207964923710, 54561107576767408522752, 2918071167402563863036269
Offset: 7
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 7):
seq(a(n), n=7..30);
-
Table[n^(n-14) * (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^6 + 51*n^5 + 1385*n^4 + 24885*n^3 + 303766*n^2 + 2333976*n + 8648640)/46080,{n,7,25}] (* Vaclav Kotesovec, Sep 06 2014 *)
A240685
Number of forests with n labeled nodes and 8 trees.
Original entry on oeis.org
1, 36, 990, 26070, 705375, 20151846, 614506893, 20073049425, 702495121185, 26300384653400, 1050925859466912, 44702294310795888, 2018603140944000000, 96508616036970572820, 4872478522317533107200, 259140537891648535707618, 14485018396686799073181696
Offset: 8
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 8):
seq(a(n), n=8..30);
-
Table[n^(n-16) * (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^7 + 63*n^6 + 2135*n^5 + 49245*n^4 + 816256*n^3 + 9527868*n^2 + 71254800*n + 259459200)/645120,{n,8,25}] (* Vaclav Kotesovec, Sep 06 2014 *)
A240686
Number of forests with n labeled nodes and 9 trees.
Original entry on oeis.org
1, 45, 1485, 45540, 1402830, 44837793, 1508782275, 53789959080, 2036262886515, 81857181636945, 3490649483399793, 157637380245930000, 7524305274666328785, 378816067488484478160, 20074256751067210380645, 1117410784286881766178816, 65207052558569641113281250
Offset: 9
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 9):
seq(a(n), n=9..30);
-
Table[n^(n-18) * (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^8 + 76*n^7 + 3122*n^6 + 88760*n^5 + 1873921*n^4 + 29555596*n^3 + 334746252*n^2 + 2455095600*n + 8821612800)/10321920,{n,9,30}] (* Vaclav Kotesovec, Sep 06 2014 *)
A240687
Number of forests with n labeled nodes and 10 trees.
Original entry on oeis.org
1, 55, 2145, 75790, 2637635, 93783690, 3467403940, 134463763720, 5491244257785, 236503301350745, 10742799174110575, 514243815022230930, 25908948794088640280, 1371861202568610407885, 76216658109172817448960, 4435598473883166992187500, 269963484584876515488140800
Offset: 10
-
T:= proc(n, m) option remember; `if`(n<0, 0, `if`(n=m, 1,
`if`(m<1 or m>n, 0, add(binomial(n-1, j-1)*j^(j-2)*
T(n-j, m-1), j=1..n-m+1))))
end:
a:= n-> T(n, 10):
seq(a(n), n=10..30);
-
Table[n^(n-20) * (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^9 + 90*n^8 + 4386*n^7 + 149436*n^6 + 3859401*n^5 + 77149170*n^4 + 1176873076*n^3 + 13044397176*n^2 + 94273812000*n + 335221286400)/185794560,{n,10,30}] (* Vaclav Kotesovec, Sep 06 2014 *)
A343805
T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 + V))/(4*t))*sqrt(1 + V)) where V = W(-2*t*x) and W denotes the Lambert function. Table read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 4, 7, 1, 9, 39, 87, 1, 16, 126, 608, 1553, 1, 25, 310, 2470, 12985, 36145, 1, 36, 645, 7560, 62595, 351252, 1037367, 1, 49, 1197, 19285, 225715, 1946259, 11481631, 35402983, 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 4, 7;
[3] 1, 9, 39, 87;
[4] 1, 16, 126, 608, 1553;
[5] 1, 25, 310, 2470, 12985, 36145;
[6] 1, 36, 645, 7560, 62595, 351252, 1037367;
[7] 1, 49, 1197, 19285, 225715, 1946259, 11481631, 35402983;
[8] 1, 64, 2044, 43232, 673190, 8011136, 71657404, 439552864, 1400424097;
-
alias(W = LambertW):
EhrB := exp(-W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
ser := series(EhrB, x, 10): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k=0..n): seq(T(n), n = 0..9);
-
P := ProductLog[-2 t x]; gf := 1/(E^((P (2 + P))/(4 t)) Sqrt[1 + P]);
ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];
Table[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten
A343806
T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 + 2*t + V))/(4*t))*sqrt(1 + V)) where V = W(-2*t*x) and W denotes the Lambert function. Table read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
1, 1, 2, 1, 6, 14, 1, 12, 66, 172, 1, 20, 192, 1080, 3036, 1, 30, 440, 4040, 23580, 69976, 1, 42, 870, 11600, 106620, 644568, 1991656, 1, 56, 1554, 28140, 364140, 3396960, 21170520, 67484880, 1, 72, 2576, 60592, 1037400, 13362272, 126973504, 811924032, 2652878864
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 2;
[2] 1, 6, 14;
[3] 1, 12, 66, 172;
[4] 1, 20, 192, 1080, 3036;
[5] 1, 30, 440, 4040, 23580, 69976;
[6] 1, 42, 870, 11600, 106620, 644568, 1991656;
[7] 1, 56, 1554, 28140, 364140, 3396960, 21170520, 67484880;
[8] 1, 72, 2576, 60592, 1037400, 13362272, 126973504, 811924032, 2652878864;
-
alias(W = LambertW):
EhrC := exp(-(t+1)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t))/sqrt(1+W(-2*t*x)):
ser := series(EhrC, x, 10): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k=0..n): seq(T(n), n = 0..8);
-
P := ProductLog[-2 t x]; gf := 1/(E^((P (2 + 2 t + P))/(4 t)) Sqrt[1 + P]);
ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];
Table[CoefficientList[cx[n], t], {n, 0, 8}] // Flatten
A343807
T(n, k) = [x^k] n! [t^n] 1/(exp((V*(2 - 2*t + V))/(4*t))*sqrt(1 + V)) where V = W(-2*t*x) and W denotes the Lambert function. Table read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
1, 1, 0, 1, 2, 2, 1, 6, 18, 32, 1, 12, 72, 280, 636, 1, 20, 200, 1320, 6060, 15744, 1, 30, 450, 4480, 32460, 166536, 470680, 1, 42, 882, 12320, 127260, 996408, 5526136, 16542336, 1, 56, 1568, 29232, 405720, 4384800, 36529920, 214436160, 669165840
Offset: 0
[0] 1;
[1] 1, 0;
[2] 1, 2, 2;
[3] 1, 6, 18, 32;
[4] 1, 12, 72, 280, 636;
[5] 1, 20, 200, 1320, 6060, 15744;
[6] 1, 30, 450, 4480, 32460, 166536, 470680;
[7] 1, 42, 882, 12320, 127260, 996408, 5526136, 16542336;
[8] 1, 56, 1568, 29232, 405720, 4384800, 36529920, 214436160, 669165840;
-
alias(W = LambertW):
EhrD := exp(-(1-t)*W(-2*t*x)/(2*t) - W(-2*t*x)^2/(4*t)) / sqrt(1+W(-2*t*x)):
ser := series(EhrD, x, 10): cx := n -> n!*coeff(ser, x, n):
T := n -> seq(coeff(cx(n), t, k), k = 0..n): seq(T(n), n = 0..8);
-
P := ProductLog[-2 t x]; gf := 1/(E^((P (2 - 2 t + P))/(4 t)) Sqrt[1 + P]);
ser := Series[gf, {x, 0, 10}]; cx[n_] := n! Coefficient[ser, x, n];
Table[If[n == 1, {1, 0}, CoefficientList[cx[n], t]], {n, 0, 8}] // Flatten
A143899
Triangle read by rows: T(n,k)=number of simple graphs on n labeled nodes with k edges containing at least one cycle subgraph, n>=3, 3<=k<=C(n,2).
Original entry on oeis.org
1, 4, 15, 6, 1, 10, 85, 252, 210, 120, 45, 10, 1, 20, 285, 1707, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 35, 735, 6972, 37457, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1, 56, 1610
Offset: 3
T(4,3) = 4, because 4 simple graphs on 4 labeled nodes with 3 edges contain a cycle subgraph:
..1-2...1-2...1.2...1.2..
..|/.....\|...|\...../|..
..3.4...3.4...3-4...3-4..
Triangle begins:
1;
4, 15, 6, 1;
10, 85, 252, 210, 120, 45, 10, 1;
20, 285, 1707, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105, 15, 1;
-
B:= proc(n) option remember; if n=0 then 0 else B(n-1) +n^(n-1) *x^n/n! fi end: BB:= proc(n) option remember; expand (B(n) -B(n)^2/2) end: f:= proc(k) option remember; if k=0 then 1 else unapply (f(k-1)(x) +x^k/k!, x) fi end: A:= proc(n,k) option remember; series(f(k)(BB(n)), x,n+1) end: aa:= (n,k)-> coeff (A(n,k), x,n) *n!: b:= (n,k)-> if k>=n then 0 else aa(n,n-k) -aa(n,n-k-1) fi: T:= (n,k)-> product (n*(n-1)/2-j, j=0..k-1)/k! -b(n,k): seq (seq (T(n,k), k=3..n*(n-1)/2), n=3..8);
-
(* t = A138464 *) t[0, 0] = 1; t[n_, k_] /; (0 <= k <= n-1) := t[n, k] = Sum[(i+1)^(i-1)*Binomial[n-1, i]*t[n-i-1, k-i], {i, 0, k}]; t[, ] = 0; T[n_, k_] := Binomial[n*(n-1)/2, k]-t[n, k]; Table[Table[T[n, k], {k, 3, n*(n-1)/2}], {n, 3, 8}] // Flatten (* Jean-François Alcover, Feb 14 2014 *)
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